
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
$S$ is the set of all non-negative integers. Find all functions \( f: S \rightarrow S \), \( g: S \rightarrow S \), and \( h: S \rightarrow S \) that satisfy the following two conditions:
1. For any \( m, n \in S \), \( f(m+n) = g(m) + h(n) + 2mn \).
2. \( g(1) = h(1) = 1 \).

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