
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 $f(x) = ax - \ln x$, $g(x) = \ln(x^2 - 2x + a)$,

(1) If $a=0$, find the zero of $F(x) = f(x) + g(x)$;

(2) Let proposition P be: $f(x)$ is monotonically decreasing in the interval $\left[\frac{1}{4}, \frac{1}{2}\right]$, and proposition Q be: the domain of $g(x)$ is $\mathbb{R}$. If $P \land Q$ is true, find the range of $a$.

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