
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 $f(x)=\frac{a\ln x}{x}+b$ has a tangent line equation $x-y-1=0$ at the point $(1,0)$.

(1) Find $a$ and $b$;
(2) Find the maximum value of $f(x)$;
(3) Let $c > 0$ and $c\neq 1$, if the function $g(x)=\log_c x-x$ has at least one zero point, find the maximum value of $c$.

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