
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 \( S \) be a set with 35 elements, and \( F \) be a collection of mappings from \( S \) to itself. For a positive integer \( k \), \( F \) is said to have the property \( P(k) \) if for any \( x, y \in S \), there exist \( k \) mappings \( f_1, f_2, \ldots, f_k \) from \( F \) (not necessarily distinct) such that
\[
f_k \left( \cdots \left( f_2 \left( f_1(x) \right) \right) \cdots \right) = f_k \left( \cdots \left( f_2 \left( f_1(y) \right) \right) \cdots \right).
\]
Find the smallest positive integer \( m \) such that all sets \( F \) with the property \( P(2019) \) also have the property \( P(m) \).

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