
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 set of mappings from $S$ to itself. For a positive integer $k$, we say that $F$ has property $P(k)$ if for any $x, y \in S$, there exist $k$ mappings $f_{1}, f_{2}, \dots, f_{k}$ in $F$ (possibly the same) such that
$$
f_{k} \left( \cdots f_{2} \left( f_{1}(x) \right) \cdots \right) = f_{k} \left( \cdots f_{2} \left( f_{1}(y) \right) \cdots \right).
$$

Find the smallest positive integer $m$ so that all $F$ with property $P(2019)$ also have property $P(m)$.

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