
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  $f : Z_{\ge 0} \to Z_{\ge 0}$  satisfy the functional equation   $$ f(m^2 + n^2) =(f(m) - f(n))^2 + f(2mn) $$   for all nonnegative integers  $m, n$ . If  $8f(0) + 9f(1) = 2006$ , compute  $f(0)$ .

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