
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 a quadratic function $f\left(x\right)=ax^{2}+bx+c$, and a set $M=\{x|ax^{2}+bx+c \gt 0\}$, where $a\neq 0$, $b$, $c\in R$.  
$(1)$ If $M=\{x|-\frac{1}{2}＜x＜2\}$ and $f\left(0\right)=2$, find the analytical expression of $f\left(x\right)$;  
$(2)$ If $a \gt 0$, $b \gt 0$, $c \gt 0$, and $M=\left(-\infty ,m\right)\cup \left(m,+\infty \right)\left(m \lt 0\right)$, find the minimum value of $\frac{b}{c}-2m$.

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