
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 $f(x)=ae^{2x}-be^{-2x}-cx$ $(a,b,c \in \mathbb{R})$, its third derivative $f'''(x)$ is an even function, and the slope of the tangent line to the curve $y=f(x)$ at the point $(0,f(0))$ is $4-c$.

$(1)$ Determine the values of $a,b$;

$(2)$ If $c=3$, discuss the monotonicity of $f(x)$;

$(3)$ If $f(x)$ has extreme values, find the range of values for $c$.

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