
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 $f(x)=ax^{3}-x^{2}+bx$ ($a,b\in\mathbb{R}$, $f'(x)$ is its derivative), and $f(x)$ has a local minimum of $-9$ when $x=3$  
(Ⅰ) Find the interval of monotonic decrease for $f(x)$;  
(Ⅱ) If the inequality $f'(x) > k(x\ln x-1)-6x-4$ ($k$ is a positive integer) holds for any positive real number $x$, find the maximum value of $k$. (The solution process may refer to the following data: $\ln 7\approx1.95$, $\ln 8\approx2.08$)

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