
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 \( f(x) = \ln \left(\mathrm{e}^{x} + a\right) \) where \( a > 0 \):
1. Find the inverse function \( y = f^{-1}(x) \) and the derivative \( f'(x) \) of the function \( f(x) \).
2. Suppose for any \( x \in [\ln(3a), \ln(4a)] \), the inequality \( \left| m - f^{-1}(x) \right| + \ln \left[ f'(x) \right] < 0 \) holds. Determine the range of the real number \( m \).

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