
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 $f(x)=a\ln x+\frac{2x+1}{x} \ (a\in \mathbb{R})$ has a tangent line at $x=2$ that is perpendicular to the line $4x+y=0$,

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

(Ⅱ) If there exists $x\in(1,+\infty)$ such that $f(x) < \frac{m(x-1)+2}{x} \ (m\in \mathbb{Z})$ holds, find the minimum value of $m$.

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