
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 that $a$ is a real number greater than $0$, and the function $f(x)=x^{2}(x-a)$.
(1) If the tangent line of the curve $y=f(x)$ at the point $(2,f(2))$ is parallel to the $X$ axis, find the value of $a$;
(2) Find the minimum value of $f(x)$ on the interval $[0,2]$;
(3) Under the condition of (1), if $g(x)=f(x)+ \frac {m}{x-1}$ is an increasing function on $[3,+∞)$, find the maximum value of the real number $m$.

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