
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 $f(x)=x^{3}+3ax^{2}+3bx+c$, it has an extreme value at $x=2$, and its graph's tangent line at $x=1$ is parallel to the line $6x+2y+5=0$.
(1) Find the function's monotonic intervals.
(2) Find the difference between the function's maximum and minimum values.
(3) If $f(x) > 1-4c^{2}$ holds true for $x\in[1,3]$, find the range of real number $c$.

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