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Contest: Task: Related: TaskD TaskF

Score : $1600$ points

Problem Statement

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 $y_i$ ($x_i < y_i$).

pic1-small-2acea94b.png

Each balancer must be in one of two states: up or down.

Consider 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:

  • 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.
  • 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.
  • Otherwise, the token doesn't change the wire it's moving along.

Let 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.

A 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.

You are given an integer $T$ ($1 \le T \le 2$). Answer the following question:

  • If $T = 1$, find any uniforming state of the network or determine that it doesn't exist.
  • If $T = 2$, find any non-uniforming state of the network or determine that it doesn't exist.

Note that if you answer just one kind of questions correctly, you can get partial score.

Constraints

  • $2 \leq N \leq 50000$
  • $1 \leq M \leq 100000$
  • $1 \leq T \leq 2$
  • $1 \leq x_i < y_i \leq N$
  • All input values are integers.

Partial Score

  • $800$ points will be awarded for passing the testset satisfying $T = 1$.
  • $800$ points will be awarded for passing the testset satisfying $T = 2$.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $T$
$x_1$ $y_1$
$x_2$ $y_2$
$:$
$x_M$ $y_M$

Output

Print any uniforming state of the given network if $T = 1$, or any non-uniforming state if $T = 2$. If the required state doesn't exist, output -1.

Sample Input 1

4 5 1
1 3
2 4
1 2
3 4
2 3

Sample Output 1

^^^^^

This state is uniforming: regardless of the starting wire, the token ends up at wire $1$.

pic2-small-2acea94b.png

Sample Input 2

4 5 2
1 3
2 4
1 2
3 4
2 3

Sample Output 2

v^^^^

This state is non-uniforming: depending on the starting wire, the token might end up at wire $1$ or wire $2$.

pic3final-small-2acea94b.png

Sample Input 3

3 1 1
1 2

Sample Output 3

-1

A uniforming state doesn't exist.

Sample Input 4

2 1 2
1 2

Sample Output 4

-1

A non-uniforming state doesn't exist.