
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 functions $f(x) = e^x$ (where $e$ is the base of the natural logarithm) and $g(x) = \ln \frac{x}{2} + \frac{1}{2}$, for any $m > 0$, there always exist $a$ and $b$ such that $f(a) = f(b) = m$. Find the minimum value of $b - a$.

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