
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 the set of all real values of \( x \) with \( 0 < x < \pi \) such that \( \sin x \), \( |\cos x| \), and \( |\tan x| \) form the side lengths (in some order) of a right triangle. Compute the sum of \( \tan^2 x \) over all \( x \) in \( S \).

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