{"problem":{"name":"Balancing Network","description":{"content":"A _balancing network_ is an abstract device built up of $N$ wires, thought of as running from left to right, and $M$ _balancers_ that connect pairs of wires. The wires are numbered from $1$ to $N$ fro","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"agc041_e"},"statements":[{"statement_type":"Markdown","content":"A _balancing network_ is an abstract device built up of $N$ wires, thought of as running from left to right, and $M$ _balancers_ that connect pairs of wires. The wires are numbered from $1$ to $N$ from top to bottom, and the balancers are numbered from $1$ to $M$ from left to right. Balancer $i$ connects wires $x_i$ and $y_i$ ($x_i < y_i$).\n\n![image](https://img.atcoder.jp/agc041/pic1-small-2acea94b.png)\n\nEach balancer must be in one of two states: _up_ or _down_.\nConsider a token that starts moving to the right along some wire at a point to the left of all balancers. During the process, the token passes through each balancer exactly once. Whenever the token encounters balancer $i$, the following happens:\n\n*   If the token is moving along wire $x_i$ and balancer $i$ is in the down state, the token moves down to wire $y_i$ and continues moving to the right.\n*   If the token is moving along wire $y_i$ and balancer $i$ is in the up state, the token moves up to wire $x_i$ and continues moving to the right.\n*   Otherwise, the token doesn't change the wire it's moving along.\n\nLet a state of the balancing network be a string of length $M$, describing the states of all balancers. The $i$\\-th character is `^` if balancer $i$ is in the up state, and `v` if balancer $i$ is in the down state.\nA state of the balancing network is called _uniforming_ if a wire $w$ exists such that, regardless of the starting wire, the token will always end up at wire $w$ and run to infinity along it. Any other state is called _non-uniforming_.\nYou are given an integer $T$ ($1 \\le T \\le 2$). Answer the following question:\n\n*   If $T = 1$, find any uniforming state of the network or determine that it doesn't exist.\n*   If $T = 2$, find any non-uniforming state of the network or determine that it doesn't exist.\n\nNote that if you answer just one kind of questions correctly, you can get partial score.\n\n## Constraints\n\n*   $2 \\leq N \\leq 50000$\n*   $1 \\leq M \\leq 100000$\n*   $1 \\leq T \\leq 2$\n*   $1 \\leq x_i < y_i \\leq N$\n*   All input values are integers.\n\n## Input\n\nInput is given from Standard Input in the following format:\n\n$N$ $M$ $T$\n$x_1$ $y_1$\n$x_2$ $y_2$\n$:$\n$x_M$ $y_M$\n\n## Partial Score\n\n*   $800$ points will be awarded for passing the testset satisfying $T = 1$.\n*   $800$ points will be awarded for passing the testset satisfying $T = 2$.\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"agc041_e","tags":[],"sample_group":[["4 5 1\n1 3\n2 4\n1 2\n3 4\n2 3","^^^^^\n\nThis state is uniforming: regardless of the starting wire, the token ends up at wire $1$.\n\n![image](https://img.atcoder.jp/agc041/pic2-small-2acea94b.png)"],["4 5 2\n1 3\n2 4\n1 2\n3 4\n2 3","v^^^^\n\nThis state is non-uniforming: depending on the starting wire, the token might end up at wire $1$ or wire $2$.\n\n![image](https://img.atcoder.jp/agc041/pic3final-small-2acea94b.png)"],["3 1 1\n1 2","\\-1\n\nA uniforming state doesn't exist."],["2 1 2\n1 2","\\-1\n\nA non-uniforming state doesn't exist."]],"created_at":"2026-03-03 11:01:14"}}