
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
Consider all functions  $f: \mathbb{Z} \rightarrow \mathbb{Z}$  satisfying \[f(f(x) + 2x + 20) = 15. \]

Call an integer  $n$   $\textit{good}$  if  $f(n)$  can take any integer value. In other words, if we fix  $n$ , for any integer  $m$ , there exists a function  $f$  such that  $f(n) = m.$  Find the sum of all good integers  $x.$   

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