
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 \( a, b \neq 0 \), and 
\[ \frac{\sin^4 x}{a^2} + \frac{\cos^4 x}{b^2} = \frac{1}{a^2 + b^2}. \]
Find the value of \( \frac{\sin^{100} x}{a^{100}} + \frac{\cos^{100} x}{b^{100}} \).

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