
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
If a function $f(x)$ is increasing in its definition domain $D$ on a certain interval $I$, and $F(x) = \frac{f(x)}{x}$ is also increasing on $I$, then $y = f(x)$ is called a "perfect function" on $I$. Given that $g(x) = e^x + x - \ln(x) + 1$, if the function $g(x)$ is a "perfect function" on the interval $\left[\frac{m}{2}, +\infty\right)$, the minimum value of the integer $m$ is $\_\_\_\_\_\_\_\_\_\_\_.$

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