
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 \( f(x)=\ln x - \frac{1}{2} a x^{2} - 2 x \) where \( a \in [-1, 0) \). Given that for all \( a \) in the given range, \( f(x) < b \) holds for the interval \( (0,1] \), determine the range of the real number \( b \).

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