
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]


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Given a function $f(x)$ with the domain $\mathbb{R}$, let $f'(x)$ denote the derivative of $f(x)$. Given that $f(1) = e$, $g(x) = f'(x) - f(x)$, $g(1) = 0$, and the derivative of $g(x)$ is always greater than zero, find the minimum value of the function $h(x) = f(x) - e^x$ (where $e \approx 2.71828$ is the base of the natural logarithm).

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