
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 \( A \) be the \( 2n \times 2n \) matrix whose diagonal elements are all \( x \) and whose off-diagonal elements \( a_{ij} = a \) for \( i + j \) even, and \( b \) for \( i + j \) odd. Find \( \lim_{x \to a} \frac{\det(A)}{(x - a)^{2n - 2}} \).

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