
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
Given a function \( f(x) \) defined on \([0,1]\) that satisfies:
1. \( f(1) = 3 \);
2. \( f(x) \geq 2 \);
3. If \( x_1 \geq 0 \), \( x_2 \geq 0 \), and \( x_1 + x_2 \leq 1 \), then \( f(x_1 + x_2) \geq f(x_1) + f(x_2) - 2 \).

(1) Find the maximum and minimum values of the function \( f(x) \).

(2) Compare \( f\left( \frac{1}{2^*} \right) \) with \( \frac{1}{2^*} + 2 \) where \( n \in \mathbf{N}^{+} \).

(3) Someone found that when \( x = \frac{1}{2^n} \) (\( n \in \mathbf{N} \)), \( f(x) < 2x + 2 \). Based on this, they propose the conjecture: For any \( x \in (0,1] \), \( f(x) < 2x + 2 \). Determine if this conjecture is correct and provide a justification.

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