
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 $f(x)=x^{2}-a\ln x$ and $g(x)=(a-2)x$.  
$(1)$ Find the monotonic interval of the function $f(x)$.  
$(2)$ If the function $F(x)=f(x)-g(x)$ has two zeros $x_{1}$ and $x_{2}$, find the smallest positive integer value of $a$ that satisfies the condition.

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