
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 $f(x) = (x - 2)\ln x + 2x - 3$, where $x \geq 1$.

(1) 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 $a$. (You may need the following data: $\ln 1.59 \approx 0.46$, $\ln 1.60 \approx 0.47$; $\frac{400}{41} \approx 9.76$)

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