
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  $0\le a_0\le a_1\le a_2\le \ldots$  of real numbers such that
\[a_{m^2+n^2}=a_m^2+a_n^2 \]
for all integers  $m,n\ge 0$ .

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