
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 \( S = \{0, 1, 2, 3, \ldots\} \) be the set of all non-negative integers. Find all functions \( f \) defined on \( S \) that take values in \( S \), and satisfy the following condition:
$$
f(m + f(n)) = f(f(m)) + f(n)
$$
for all \( m, n \in S \).

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