
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 $f(x) = \ln{x}$, and $g(x) = \frac{3}{2} - \frac{a}{x}$ (where $x$ is a positive real number),
1. When $a=1$, find the minimum value of the function $\varphi(x) = f(x) - g(x)$ for $x \in [4, +\infty)$.
2. If the equation $e^{2f(x)} = g(x)$ (where $e = 2.71828\ldots$) has a solution on the interval $\left[\frac{1}{2}, 1\right]$, find the range of the real number $a$.

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