
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 $f\left(x\right)={x}^{2}-2a\mathrm{ln}x \left(a\in \mathbb{R}\right)$.

(1) Discuss the monotonicity of the function $f\left(x\right)$;

(2) If $a>0$, suppose the function $g\left(x\right)=f\left(x\right)-2ax$ has a unique zero point, find the value of $a$.

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