
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 $$f(x)=\ln x- \frac {1}{2}ax^{2}$$ (where $a\in\mathbb{R}$),
1. If the tangent line to $f(x)$ at the point $(2, f(2))$ is perpendicular to the line $2x+y+2=0$, find the value of the real number $a$;
2. Determine the intervals of monotonicity for the function $f(x)$;
3. Discuss the number of zeros of the function $f(x)$ in the interval $[1, e^2]$.

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