
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 \( a \) and \( b \) be non-zero real numbers and \( x \in \mathbb{R} \). If
\[ 
\frac{\sin^{4} x}{a^{2}} + \frac{\cos^{4} x}{b^{2}} = \frac{1}{a^{2} + b^{2}},
\]
find the value of \(\frac{\sin^{2008} x}{a^{2006}} + \frac{\cos^{2008} x}{b^{2006}}\).

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