
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
Compute the number of triples \((f, g, h)\) of permutations on \(\{1,2,3,4,5\}\) such that
\[
\begin{array}{l}
f(g(h(x))) = h(g(f(x))) = g(x), \\
g(h(f(x))) = f(h(g(x))) = h(x), \text{ and } \\
h(f(g(x))) = g(f(h(x))) = f(x)
\end{array}
\]
for all \(x \in \{1,2,3,4,5\}\).

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