
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 $f\left(x\right)=\dfrac{1}{2}x^{2}-m\ln x$ where $m\in R$.
1. When $m=2$, find the maximum and minimum values of the function $f\left(x\right)$ on $\left[1,e\right]$.
2. If the function $f\left(x\right)$ is monotonically increasing on $\left(\dfrac{1}{2},+\infty\right)$, find the range of real number $m$.

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