
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 the sequence \(a_{0}, a_{1}, a_{2}, \cdots\) satisfy:
\[ a_{0} = a_{1} = 11, \quad a_{m+n} = \frac{1}{2}(a_{2m} + a_{2n}) - (m-n)^{2}, \quad m, n \geq 0. \]

Find \(a_{45}\).

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