
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
Find all sequences  $a_1, a_2, a_3, \dots$  of real numbers such that for all positive integers  $m,n\ge 1$ , we have
\begin{align*}
a_{m+n} &= a_m+a_n - mn \text{ and} 
a_{mn} &= m^2a_n + n^2a_m + 2a_ma_n. 
\end{align*}

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