
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
The function $f(x)=x^{2}-2ax+\ln x$ ($a\in\mathbb{R}$).

(I) The tangent line to the function $y=f(x)$ at the point $(1,f(1))$ is perpendicular to the line $x-2y+1=0$. Find the value of $a$.

(II) Discuss the monotonicity of the function $f(x)$.

(III) The inequality $2x\ln x\geqslant -x^{2}+ax-3$ holds for the interval $(0,e]$. Find the range of the real number $a$.

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