
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 \( w = w_1, w_2, \ldots, w_6 \) be a permutation of the integers \(\{1, 2, \ldots, 6\}\). If there do not exist indices \(i < j < k\) such that \( w_i < w_j < w_k \) or indices \( i < j < k < l \) such that \( w_i > w_j > w_k > w_l \), then \(w\) is said to be exquisite. Find the number of exquisite permutations.

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