
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 functions $f(x)=ax-\ln x$ and $g(x)=\ln (x^{2}-2x+a)$,
(1) If $a=0$, find the zeros of the function $F(x)=f(x)+g(x)$;
(2) Let proposition $P$: $f(x)$ is monotonically decreasing on $\left[ \frac{1}{4}, \frac{1}{2}\right]$, and proposition $q$: the domain of $g(x)$ is $\mathbb{R}$. If the compound proposition $p∧q$ is true, find the range of $a$.

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