
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 \( f(x) = \ln(1 + ax) - x^2 \) where \( a > 0 \) and \( x \in (0, 1] \):
1. Determine the intervals in which the function \( f(x) \) is monotonic.
2. Given that the inequality \( \frac{1}{n^2} + \lambda \geqslant \ln \left(1 + \frac{2}{n}\right) \) holds for all positive integers \( n \), find the range of the real number \( \lambda \).

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