
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
The function $f$ maps nonnegative integers to real numbers, given that $f(1) = 4,$ and for all nonnegative integers $m \geq n,$
\[ f(m + n) + f(m - n) = \frac{f(2m) + f(2n)}{2}. \]
Find the sum of all possible values of $f(10)$.

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