
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 $f(x)=\frac{1}{2}x^2-(2a+2)x+(2a+1)\ln x$,
(1) Discuss the monotonicity of the function $y=f(x)$;
(2) For any $a \in \left[\frac{1}{2}, 2\right]$, $x_1, x_2 \in [1,2]$ ($x_1 \neq x_2$), there always exists $|f(x_1)-f(x_2)| < \lambda \bigg|\frac{1}{x_1}-\frac{1}{x_2}\bigg|$, find the range of the positive real number $\lambda$.

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