
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 $f(x)=x+a\ln x$, where $a\in R$.
(1) Find the monotonic intervals of the function $f(x)$;
(2) When $x\in[1,2]$, $f(x) > 0$ holds, find the range of $a$;
(3) Try to find how many tangent lines can be drawn through the point $P(1,3)$ to the curve $y=f(x)$, and explain the reason.

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