
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 $f(x)=2\ln \frac{x}{2}- \frac{3x-6}{x+1}$,

(Ⅰ) Find the intervals of monotonicity for $f(x)$;

(Ⅱ) If $g(x)=(x-t)^{2}+(\ln x-at)^{2}$, and for any $x_{1}\in(1,+\infty)$, there exists $t\in(-\infty,+\infty)$, $x_{2}\in(0,+\infty)$, such that $f(x_{1})\geqslant g(x_{2})$ holds, find the range of the real number $a$.

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