
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 sequence \( a_{0}, a_{1}, a_{2}, \cdots \) satisfying \( a_{0} = a_{1} = 11 \) and \( a_{m+n} = \frac{1}{2}\left(a_{2m}+a_{2n}\right) - (m-n)^2 \) for \( m, n \geq 0 \), find \( a_{45} \).

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