
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 the function \( f(x) = \ln (x+1) + \frac{2}{x+1} + ax - 2 \) (where \( a > 0 \)).

(1) When \( a = 1 \), find the minimum value of \( f(x) \).

(2) If \( f(x) \geq 0 \) holds for \( x \in [0, 2] \), find the range of the real number \( a \).

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