
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 \( x \) for \( 0 < x < \frac{\pi}{4} \) such that \( \sin x \), \( \cos x \), and \( \tan x \) form the side lengths (in some order) of a right triangle. Find and sum all values of \( \sin^2 x \) over all \( x \) in \( S \).

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