
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 $f(x)=ax^{2}+bx+c$, where $a\in\mathbb{N}^{*}$, $b\in\mathbb{N}$, $c\in\mathbb{Z}$.

$(1)$ If $b > 2a$, and $f(\sin x)$ ($x\in\mathbb{R}$) has a maximum value of $2$ and a minimum value of $-4$, find the minimum value of the function $f(x)$.

$(2)$ For the $f(x)$ in $(1)$, if for any $m\in[-4,1]$, it always holds that $f(x)\geqslant 2x^{2}-mx-14$, find the range of values for $x$.

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