
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 $f(x)=(x-2)\ln x+2x-3$, $x\geqslant 1$.  
$(1)$ Try to determine the number of zeros of the function $f(x)$.  
$(2)$ If the function $g(x)=(x-a)\ln x+ \frac {a(x-1)}{x}$ is increasing on $[1,+\infty)$, find the maximum integer value of $a$. (Data you may need: $\ln 1.59\approx0.46$; $\ln 1.60\approx0.47$; $ \frac {400}{41}\approx9.76)$

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