
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 \( 2 \mathbf{Z} + 1 \) denote the set of all odd numbers. Find all functions \( f: \mathbf{Z} \rightarrow 2 \mathbf{Z} + 1 \) such that for any \( x, y \in \mathbf{Z} \), the following equation holds:
\[
f(x + f(x) + y) + f(x - f(x) - y) = f(x + y) + f(x - y)
\]

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