
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Suppose there are  $160$  pigeons and  $n$  holes. The  $1$ st pigeon flies to the  $1$ st hole, the  $2$ nd pigeon flies to the  $4$ th hole, and so on, such that the  $i$ th pigeon flies to the  $(i^2\text{ mod }n)$ th hole, where  $k\text{ mod }n$  is the remainder when  $k$  is divided by  $n$ . What is minimum  $n$  such that there is at most one pigeon per hole?

*Proposed by Christina Yao*

Present the answer in LaTex format: \boxed{Your answer}
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