
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 $f(x)=ax^3+bx^2-5x+\frac{1}{3}$ with its derivative $f'(x)$, the function attains a maximum value of 3 at $x=-1$.
(1) Find the interval where the function is monotonically decreasing.
(2) If for any positive real number $x$, $f'(x)>k(x\ln x-2)-5x-9$ holds true (with $k$ being a positive integer), find the maximum value of $k$.
(Note: $\ln 2\approx 0.693$, $\ln 3\approx 1.099$, $\ln 5\approx 1.609$, $\ln 7\approx 1.946$)

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