
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 $f(x)=\ln x+1+\frac{2a}{x}$. The tangent line to the function at the point $(a,f(a))$ passes through the point $(0,4)$.
(I) Find the value of the real number $a$ and the monotonic intervals of the function $f(x)$;
(II) If the integer $k$ satisfies $2f(x) > k(1-\frac{1}{x})$ for all $x\in(1,+\infty)$, find the maximum value of $k$.

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