
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 $f(x) = (2-a)(x-1) - 2\ln x$, $g(x) = xe^{1-x}$ ($a \in \mathbb{R}$, $e$ is the base of the natural logarithm),
1. When $a=1$, find the intervals of monotonicity for $f(x)$;
2. If the function $f(x)$ has no zeros in the interval $(0, \frac{1}{2})$, find the minimum value of $a$.

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