
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
Find \( a_{0} \) and \( a_{l, k} \) for \( 1 \leq l < k \leq n \) such that for any \( x \in \mathbb{R} \) (where \( \sin x \neq 0 \)), the following equation holds:
\[ 
\frac{\sin^{2}(nx)}{\sin^{2} x} = a_{0} + \sum_{1 \leq l < k \leq n} a_{l, k} \cdot \cos 2(k-l)x.
\]

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