
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 the set of mappings from \( S \) to itself. For a positive integer \( k \), the set \( F \) is said to possess property \( P(k) \) if for any \( x, y \in S \), there exist \( k \) mappings \( f_{1}, f_{2}, \cdots, f_{k} \) from \( F \) (which may not be 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 \( F \) with property \( P(2019) \) also possess property \( P(m) \).

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