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

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