
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
For all non-negative integers \( m \) and \( n \) (with \( m \geq n \)), the sequence \( a_0, a_1, \cdots \) satisfies
\[ a_{m+n} + a_{m-n} + 2m - 2n - 1 = \frac{1}{2}(a_{2m} + a_{2n}). \]

If \( a_1 = 0 \) and \( a_p = 2019^2 \), find the value of \( p \).

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