
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
Let the function $f(x)=(x^2-2x)\ln x+(a-\frac{1}{2})x^2+2(1-a)x+a$, where $a\in \mathbb{R}$.

$(1)$ Discuss the monotonicity of the function $f(x)$;

$(2)$ When $a < -2$, discuss the number of zeros of $f(x)$.

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