
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 \( n \) 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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