
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 $f(x)=\frac{1}{3}x^{3}+ax+b$ $(a,b\in\mathbb{R})$ has a local minimum at $x=2$ with the value of $-\frac{4}{3}$.  
$(1)$ Find the intervals of increase for the function $f(x)$;  
$(2)$ If $f(x)\leqslant m^{2}+m+\frac{10}{3}$ holds for all $x\in[-4,3]$, find the range of the real number $m$.

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