
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 \( k \in \mathbb{N}^* \). Suppose that all positive integers are colored using \( k \) different colors, and there exists a function \( f: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+ \) satisfying:
1. For positive integers \( m \) and \( n \) of the same color (they can be the same), \( f(m+n) = f(m) + f(n) \);
2. There exist positive integers \( m \) and \( n \) (they can be the same) such that \( f(m+n) \neq f(m) + f(n) \).

Find the minimum value of \( k \).

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