
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 $f(x)= \frac {lnx+1}{x}$ and $g(x)=x\, lnx- \frac {1}{2}ax^{2}+ \frac {a}{2}$ (where $a\in\mathbb{R}$),

1. Discuss the monotonicity of the function $f(x)$;
2. If the function $g(x)$ has two extreme points, determine the number of zeros of the function $g(x)$.

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