
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 $f(x)=\frac{1}{3}x^3-bx+c$, where $b,c\in\mathbb{R}$:
1. If the equation of the tangent line to the function $f(x)$ at the point $(1,f(1))$ is $y=2x+1$, find the values of $b$ and $c$.
2. If $b=1$, and the function $f(x)$ has a unique zero in the interval $(0,2)$, find the range of values for $c$.
3. If for any $x\_1, x\_2\in[-1,1]$, we have $|f(x\_1)-f(x\_2)|\leq\frac{4}{3}$, find the range of values for $b$.

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