
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) = lnx + 1 and g(x) = $2e^{x-\frac{1}{2}}$, find the minimum value of m - n if f(m) = g(n) holds true. The options are:
A) $\frac{1}{2} + \ln{2}$
B) $e - 2$
C) $\ln{2} - \frac{1}{2}$
D) $\sqrt{e} - \frac{1}{2}$

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