{"trustable":true,"prependHtml":"\u003cstyle type\u003d\"text/css\"\u003e\n section pre {\n display: block;\n padding: 9.5px;\n margin: 0 0 10px;\n font-size: 13px;\n line-height: 1.42857143;\n word-break: break-all;\n word-wrap: break-word;\n color: #333;\n background: rgba(255, 255, 255, 0.5);\n border: 1px solid #ccc;\n border-radius: 6px;\n }\n\u003c/style\u003e\n\u003cscript\u003e\n window.katexOptions \u003d {\n delimiters: [\n {left: \u0027\\\\(\u0027, right: \u0027\\\\)\u0027, display: false},\n {left: \u0027\\\\[\u0027, right: \u0027\\\\]\u0027, display: true}\n ]\n };\n\u003c/script\u003e\n","sections":[{"title":"Problem Statement","value":{"format":"HTML","content":"\r\n\u003csection\u003e\r\n\u003cp\u003e\u003cstrong\u003eBeware the special input format and the smaller memory limit than usual.\u003c/strong\u003e\u003c/p\u003e\r\n\u003cp\u003eThere is an undirected graph with vertices \u003cvar\u003e\\(1, 2, \\dots, N\\)\u003c/var\u003e, initially without edges.\u003cbr\u003e\r\nYou need to process the following \u003cvar\u003e\\(Q\\)\u003c/var\u003e queries on this graph:\u003c/p\u003e\r\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\r\n\u003cpre\u003e1 \u003cvar\u003e\\(u\\)\u003c/var\u003e \u003cvar\u003e\\(v\\)\u003c/var\u003e\r\n\u003c/pre\u003e\r\n\u003cp\u003eType \u003cvar\u003e\\(1\\)\u003c/var\u003e: Add an edge between vertices \u003cvar\u003e\\(u\\)\u003c/var\u003e and \u003cvar\u003e\\(v\\)\u003c/var\u003e.\u003cbr\u003e\r\nBefore adding the edge, \u003cvar\u003e\\(u\\)\u003c/var\u003e and \u003cvar\u003e\\(v\\)\u003c/var\u003e belong to different connected components (i.e., the graph always remains a forest).\u003c/p\u003e\r\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\r\n\u003cpre\u003e2 \u003cvar\u003e\\(u\\)\u003c/var\u003e \u003cvar\u003e\\(v\\)\u003c/var\u003e\r\n\u003c/pre\u003e\r\n\u003cp\u003eType \u003cvar\u003e\\(2\\)\u003c/var\u003e: If there is a vertex adjacent to both \u003cvar\u003e\\(u\\)\u003c/var\u003e and \u003cvar\u003e\\(v\\)\u003c/var\u003e, report its vertex number; otherwise, report \u003cvar\u003e\\(0\\)\u003c/var\u003e.\u003cbr\u003e\r\nGiven that the graph always remains a forest, the answer to this query is uniquely determined.\u003c/p\u003e\r\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\r\n\u003cp\u003eThe queries are given in an encrypted form.\u003cbr\u003e\r\nThe original query is defined by three integers \u003cvar\u003e\\(A, B, C\\)\u003c/var\u003e, and the encrypted query is given as three integers \u003cvar\u003e\\(a, b, c\\)\u003c/var\u003e.\u003cbr\u003e\r\nLet \u003cvar\u003e\\(X_k\\)\u003c/var\u003e be the answer to the \u003cvar\u003e\\(k\\)\u003c/var\u003e-th type-\u003cvar\u003e\\(2\\)\u003c/var\u003e query. Define \u003cvar\u003e\\(X_k \u003d 0\\)\u003c/var\u003e for \u003cvar\u003e\\(k \u003d 0\\)\u003c/var\u003e.\u003cbr\u003e\r\nRestore \u003cvar\u003e\\(A, B, C\\)\u003c/var\u003e from the given \u003cvar\u003e\\(a, b, c\\)\u003c/var\u003e as follows:\u003c/p\u003e\r\n\u003cul\u003e\r\n\u003cli\u003eLet \u003cvar\u003e\\(l\\)\u003c/var\u003e be the number of type-\u003cvar\u003e\\(2\\)\u003c/var\u003e queries given before this query (not counting the query itself). Then, use the following:\u003cul\u003e\r\n\u003cli\u003e\u003cvar\u003e\\(A \u003d 1 + (((a \\times (1+X_l)) \\mod 998244353) \\mod 2)\\)\u003c/var\u003e\u003c/li\u003e\r\n\u003cli\u003e\u003cvar\u003e\\(B \u003d 1 + (((b \\times (1+X_l)) \\mod 998244353) \\mod N)\\)\u003c/var\u003e\u003c/li\u003e\r\n\u003cli\u003e\u003cvar\u003e\\(C \u003d 1 + (((c \\times (1+X_l)) \\mod 998244353) \\mod N)\\)\u003c/var\u003e\u003c/li\u003e\r\n\u003c/ul\u003e\r\n\u003c/li\u003e\r\n\u003c/ul\u003e\r\n\u003c/section\u003e\r\n"}},{"title":"Constraints","value":{"format":"HTML","content":"\r\n\u003csection\u003e\r\n\u003cul\u003e\r\n\u003cli\u003eAll input values are integers.\u003c/li\u003e\r\n\u003cli\u003e\u003cvar\u003e\\(2 \\le N \\le 10^5\\)\u003c/var\u003e\u003c/li\u003e\r\n\u003cli\u003e\u003cvar\u003e\\(1 \\le Q \\le 10^5\\)\u003c/var\u003e\u003c/li\u003e\r\n\u003cli\u003e\u003cvar\u003e\\(1 \\le u \u0026lt; v \\le N\\)\u003c/var\u003e\u003c/li\u003e\r\n\u003cli\u003e\u003cvar\u003e\\(0 \\le a,b,c \u0026lt; 998244353\\)\u003c/var\u003e\u003c/li\u003e\r\n\u003c/ul\u003e\r\n\u003c/section\u003e\r\n"}},{"title":"Input","value":{"format":"HTML","content":"\r\n\u003csection\u003e\r\n\u003cp\u003eThe input is given from Standard Input in the following format:\u003c/p\u003e\r\n\u003cpre\u003e\u003cvar\u003e\\(N\\)\u003c/var\u003e \u003cvar\u003e\\(Q\\)\u003c/var\u003e\r\n\u003cvar\u003e\\(\\rm{Query}\\)\u003c/var\u003e\u003cvar\u003e\\(_1\\)\u003c/var\u003e\r\n\u003cvar\u003e\\(\\rm{Query}\\)\u003c/var\u003e\u003cvar\u003e\\(_2\\)\u003c/var\u003e\r\n\u003cvar\u003e\\(\\vdots\\)\u003c/var\u003e\r\n\u003cvar\u003e\\(\\rm{Query}\\)\u003c/var\u003e\u003cvar\u003e\\(_Q\\)\u003c/var\u003e\r\n\u003c/pre\u003e\r\n\u003c/section\u003e\r\n"}},{"title":"Output","value":{"format":"HTML","content":"\r\n\u003csection\u003e\r\n\u003cp\u003eLet \u003cvar\u003e\\(k\\)\u003c/var\u003e be the number of type-\u003cvar\u003e\\(2\\)\u003c/var\u003e queries. Print \u003cvar\u003e\\(k\\)\u003c/var\u003e lines.\u003cbr\u003e\r\nThe \u003cvar\u003e\\(i\\)\u003c/var\u003e-th line should contain the answer to the \u003cvar\u003e\\(i\\)\u003c/var\u003e-th type-\u003cvar\u003e\\(2\\)\u003c/var\u003e query.\u003c/p\u003e\r\n\u003c/section\u003e\r\n"}},{"title":"Sample 1","value":{"format":"HTML","content":"\u003ctable class\u003d\u0027vjudge_sample\u0027\u003e\n\u003cthead\u003e\n \u003ctr\u003e\n \u003cth\u003eInput\u003c/th\u003e\n \u003cth\u003eOutput\u003c/th\u003e\n \u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd\u003e\u003cpre\u003e6 12\r\n143209915 123840720 97293110\r\n89822758 207184717 689046893\r\n67757230 270920641 26993265\r\n952858464 221010240 871605818\r\n730183361 147726243 445163345\r\n963432357 295317852 195433903\r\n120904472 106195318 615645575\r\n817920568 27584394 770578609\r\n38727673 250957656 506822697\r\n139174867 566158852 412971999\r\n205467538 606353836 855642999\r\n159292205 319166257 51234344\r\n\u003c/pre\u003e\u003c/td\u003e\n \u003ctd\u003e\u003cpre\u003e0\r\n2\r\n0\r\n2\r\n6\r\n0\r\n1\r\n\u003c/pre\u003e\u003c/td\u003e\n \u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\r\n\u003csection\u003e\r\n\r\n\u003c/section\u003e\r\n\r\n\u003csection\u003e\r\n\r\n\u003cp\u003eAfter decrypting all queries, the input is as follows:\u003c/p\u003e\r\n\u003cpre\u003e6 12\r\n2 1 3\r\n1 2 6\r\n1 2 4\r\n1 1 3\r\n2 4 6\r\n2 1 4\r\n1 5 6\r\n1 1 2\r\n2 1 4\r\n2 2 5\r\n2 3 4\r\n2 2 3\r\n\u003c/pre\u003e\r\n\u003cp\u003eThis input has a \u003cvar\u003e\\(6\\)\u003c/var\u003e-vertex graph and \u003cvar\u003e\\(12\\)\u003c/var\u003e queries.\u003c/p\u003e\r\n\u003cul\u003e\r\n\u003cli\u003eThe first query is \u003ccode\u003e2 1 3\u003c/code\u003e.\u003cul\u003e\r\n\u003cli\u003eNo vertex is adjacent to both vertex \u003cvar\u003e\\(1\\)\u003c/var\u003e and vertex \u003cvar\u003e\\(3\\)\u003c/var\u003e, so report \u003cvar\u003e\\(0\\)\u003c/var\u003e.\u003c/li\u003e\r\n\u003c/ul\u003e\r\n\u003c/li\u003e\r\n\u003cli\u003eThe second query is \u003ccode\u003e1 2 6\u003c/code\u003e.\u003cul\u003e\r\n\u003cli\u003eAdd an edge between vertices \u003cvar\u003e\\(2\\)\u003c/var\u003e and \u003cvar\u003e\\(6\\)\u003c/var\u003e.\u003c/li\u003e\r\n\u003c/ul\u003e\r\n\u003c/li\u003e\r\n\u003cli\u003eThe third query is \u003ccode\u003e1 2 4\u003c/code\u003e.\u003cul\u003e\r\n\u003cli\u003eAdd an edge between vertices \u003cvar\u003e\\(2\\)\u003c/var\u003e and \u003cvar\u003e\\(4\\)\u003c/var\u003e.\u003c/li\u003e\r\n\u003c/ul\u003e\r\n\u003c/li\u003e\r\n\u003cli\u003eThe fourth query is \u003ccode\u003e1 1 3\u003c/code\u003e.\u003cul\u003e\r\n\u003cli\u003eAdd an edge between vertices \u003cvar\u003e\\(1\\)\u003c/var\u003e and \u003cvar\u003e\\(3\\)\u003c/var\u003e.\u003c/li\u003e\r\n\u003c/ul\u003e\r\n\u003c/li\u003e\r\n\u003cli\u003eThe fifth query is \u003ccode\u003e2 4 6\u003c/code\u003e.\u003cul\u003e\r\n\u003cli\u003eThe vertex adjacent to both vertices \u003cvar\u003e\\(4\\)\u003c/var\u003e and \u003cvar\u003e\\(6\\)\u003c/var\u003e is vertex \u003cvar\u003e\\(2\\)\u003c/var\u003e.\u003c/li\u003e\r\n\u003c/ul\u003e\r\n\u003c/li\u003e\r\n\u003cli\u003eThe sixth query is \u003ccode\u003e2 1 4\u003c/code\u003e.\u003cul\u003e\r\n\u003cli\u003eNo vertex is adjacent to both vertices \u003cvar\u003e\\(1\\)\u003c/var\u003e and \u003cvar\u003e\\(4\\)\u003c/var\u003e, so report \u003cvar\u003e\\(0\\)\u003c/var\u003e.\u003c/li\u003e\r\n\u003c/ul\u003e\r\n\u003c/li\u003e\r\n\u003cli\u003eThe seventh query is \u003ccode\u003e1 5 6\u003c/code\u003e.\u003cul\u003e\r\n\u003cli\u003eAdd an edge between vertices \u003cvar\u003e\\(5\\)\u003c/var\u003e and \u003cvar\u003e\\(6\\)\u003c/var\u003e.\u003c/li\u003e\r\n\u003c/ul\u003e\r\n\u003c/li\u003e\r\n\u003cli\u003eThe eighth query is \u003ccode\u003e1 1 2\u003c/code\u003e.\u003cul\u003e\r\n\u003cli\u003eAdd an edge between vertices \u003cvar\u003e\\(1\\)\u003c/var\u003e and \u003cvar\u003e\\(2\\)\u003c/var\u003e.\u003c/li\u003e\r\n\u003c/ul\u003e\r\n\u003c/li\u003e\r\n\u003cli\u003eThe ninth query is \u003ccode\u003e2 1 4\u003c/code\u003e.\u003cul\u003e\r\n\u003cli\u003eThe vertex adjacent to both vertices \u003cvar\u003e\\(1\\)\u003c/var\u003e and \u003cvar\u003e\\(4\\)\u003c/var\u003e is vertex \u003cvar\u003e\\(2\\)\u003c/var\u003e.\u003c/li\u003e\r\n\u003c/ul\u003e\r\n\u003c/li\u003e\r\n\u003cli\u003eThe tenth query is \u003ccode\u003e2 2 5\u003c/code\u003e.\u003cul\u003e\r\n\u003cli\u003eThe vertex adjacent to both vertices \u003cvar\u003e\\(2\\)\u003c/var\u003e and \u003cvar\u003e\\(5\\)\u003c/var\u003e is vertex \u003cvar\u003e\\(6\\)\u003c/var\u003e.\u003c/li\u003e\r\n\u003c/ul\u003e\r\n\u003c/li\u003e\r\n\u003cli\u003eThe eleventh query is \u003ccode\u003e2 3 4\u003c/code\u003e.\u003cul\u003e\r\n\u003cli\u003eNo vertex is adjacent to both vertices \u003cvar\u003e\\(3\\)\u003c/var\u003e and \u003cvar\u003e\\(4\\)\u003c/var\u003e, so report \u003cvar\u003e\\(0\\)\u003c/var\u003e.\u003c/li\u003e\r\n\u003c/ul\u003e\r\n\u003c/li\u003e\r\n\u003cli\u003eThe twelfth query is \u003ccode\u003e2 2 3\u003c/code\u003e.\u003cul\u003e\r\n\u003cli\u003eThe vertex adjacent to both vertices \u003cvar\u003e\\(2\\)\u003c/var\u003e and \u003cvar\u003e\\(3\\)\u003c/var\u003e is vertex \u003cvar\u003e\\(1\\)\u003c/var\u003e.\u003c/li\u003e\r\n\u003c/ul\u003e\r\n\u003c/li\u003e\r\n\u003c/ul\u003e\r\n\u003c/section\u003e\r\n"}},{"title":"Sample 2","value":{"format":"HTML","content":"\u003ctable class\u003d\u0027vjudge_sample\u0027\u003e\n\u003cthead\u003e\n \u003ctr\u003e\n \u003cth\u003eInput\u003c/th\u003e\n \u003cth\u003eOutput\u003c/th\u003e\n \u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd\u003e\u003cpre\u003e2 1\r\n377373366 41280240 33617925\r\n\u003c/pre\u003e\u003c/td\u003e\n \u003ctd\u003e\u003cpre\u003e\u003c/pre\u003e\u003c/td\u003e\n \u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\r\n\u003csection\u003e\r\n\r\n\u003c/section\u003e\r\n\r\n\u003csection\u003e\r\n\r\n\u003cp\u003eThe output may be empty.\u003c/p\u003e\u003c/section\u003e\r\n"}}]}