
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 $f(x)= \frac {1}{2}x^{2}-(2a+2)x+(2a+1)\ln x$.  
$(1)$ Find the monotonic intervals of $f(x)$;  
$(2)$ For any $a\in\left[ \frac {3}{2}, \frac {5}{2}\right]$, $x_{1}$, $x_{2}\in[1,2]$, it always holds that $|f(x_{1})-f(x_{2})|\leqslant \lambda| \frac {1}{x_{1}}- \frac {1}{x_{2}}|$, find the range of the positive real number $\lambda$.

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