
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
Let \( P_{1}(x), P_{2}(x), \ldots, P_{k}(x) \) be monic polynomials of degree 13 with integer coefficients. Suppose there are pairwise distinct positive integers \( n_{1}, n_{2}, \ldots, n_{k} \) such that, for all positive integers \( i \) and \( j \) less than or equal to \( k \), the statement " \( n_{i} \) divides \( P_{j}(m) \) for every integer \( m \) " holds if and only if \( i = j \). Compute the largest possible value of \( k \).

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