
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 $f(x)=2\ln x- \frac{1}{2}ax^{2}+(2-a)x (a\in R)$.

1. Discuss the monotonicity of the function $f(x)$;
2. If for all $x\_1$, $x\_2\in(0,+\infty)$, and $x\_1 < x\_2$, there exists a positive real number $x\_0$ such that $f(x\_2)-f(x\_1)=f'(x\_0)(x\_2-x\_1)$, try to determine the relationship between $f'( \frac{x\_1+x\_2}{2})$ and $f'(x\_0)$, and provide a proof.

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