
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 \( w, x, y, \) and \( z \) be positive real numbers such that
\[
\begin{aligned}
0 & \neq \cos w \cos x \cos y \cos z \\
2 \pi & = w + x + y + z \\
3 \tan w & = k(1 + \sec w) \\
4 \tan x & = k(1 + \sec x) \\
5 \tan y & = k(1 + \sec y) \\
6 \tan z & = k(1 + \sec z)
\end{aligned}
\]
(Here \( \sec t \) denotes \( \frac{1}{\cos t} \) when \( \cos t \neq 0 \).) Find \( k \).

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