
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 following propositions:
$(①)$ If $a > 0$, $b > 0$, and the function $f(x)=4x^{3}-ax^{2}-2bx+2$ has an extreme value at $x=1$, then the maximum value of $ab$ is $9$;
$(②)$ Given the function $f(x)=x^{2}- \cos x$, if $x_{1}$, ${x_{2}}\in \left[ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right]$ and $f(x_{1}) > f(x_{2})$, then $|x_{1}| > |x_{2}|$;
$(③)$ If $a > 2$, then the equation $x^{3}-3ax^{2}+3=0$ has exactly one root in the interval $(0,2)$;
$(④)$ The tangent line equation of the curve $y=3x-x^{3}$ at point $A(2,-2)$ is $9x+y-16=0.$
Determine the sequence number(s) of the correct proposition(s).

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