
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
Define the function $f(x)$ on $\mathbb{R}$ as $f(x)=\frac{f'(1)}{2}\cdot e^{2^{x-2}}+x^2-2f(0)x$, and $g(x)=f\left(\frac{x}{2}\right)-\frac{1}{4}x^2+(1-a)x+a$.

(Ⅰ) Find the explicit formula for the function $f(x)$;

(Ⅱ) Determine the monotonic intervals of the function $g(x)$;

(Ⅲ) If $s$, $t$, $r$ satisfy $|s-r|\leqslant |t-r|$, then $s$ is said to be closer to $r$ than $t$. When $a\geqslant 2$ and $x\geqslant 1$, compare which one is closer to $\ln x$, $\frac{e}{x}$ or $e^{x-1}+a$, and explain why.

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