
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 $f(x)=x^3+ax^2+bx$ $(x > 0)$, its graph is tangent to the x-axis at the point $(3,0)$.

$(1)$ Find the analytical expression of the function $f(x)$;

$(2)$ If $g(x)+f(x)=-6x^2+(3c+9)x$, and proposition $p$: $\exists x_1,x_2 \in [-1,1]$, $|g(x_1)-g(x_2)| > 1$ is a false proposition, find the range of the real number $c$.

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