
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
Given \( f(x) + g(x) = \sqrt{\frac{1 + \cos 2x}{1 - \sin x}} \) for \( x \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \), where \( f(x) \) is an odd function and \( g(x) \) is an even function, determine the value of \( [f(x)]^2 - [g(x)]^2 \).

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