
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 $f(x) = e^x - a(\ln x + 1)$ $(a \in \mathbb{R})$.

$(1)$ Find the equation of the tangent line to the curve $y = f(x)$ at the point $(1, f(1))$;

$(2)$ If $a > 0$, let $g(x) = f'(x)$, determine the monotonicity of $g(x)$;

$(3)$ If the function $y = f(x)$ has an extreme value on the interval $(\frac{1}{2}, 1)$, find the range of $a$.

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