
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 \( f(x) = \ln (2 + 3x) - \frac{3}{2} x^2 \), if for any \( x \in \left[ \frac{1}{6}, \frac{1}{3} \right] \), the inequality \( |a - \ln x| + \ln \left[ f'(x) + 3x \right] > 0 \) always holds, then the range of the real number \( a \) is _______.

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