
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 $f(x) = (2 - a)(x - 1) - 2 \ln x (a \in \mathbb{R})$.

(1) If the tangent line of the curve $g(x) = f(x) + x$ at the point $(1, g(1))$ passes through the point $(0, 2)$, find the monotonically decreasing interval of the function $g(x)$;

(2) If the function $y = f(x)$ has no zeros in the interval $(0, \frac{1}{2})$, find the minimum value of the real number $a$.

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