
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 that the solution set of the inequality $cx^{2}+x+b \lt 0$ with respect to $x$ is $\{x|-1＜x＜\frac{1}{2}\}$, the solution set of the inequality $bx^{2}+x+c \gt 0$ with respect to $x$ is $M$, and the solution set of the inequality $x^{2}+x \lt a^{2}-a$ with respect to $x$ is $N$, satisfying $M\cup (\complement _{R}N)=R$.  
$(1)$ Find the set $M$;  
$(2)$ Find the range of real number $a$.

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