
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 inequality $ax^{2}+5x-2 > 0$ has the solution set $\left\{ x \mid \frac{1}{2} < x < 2 \right\}$.

$(1)$ Find the value of the real number $a$;

$(2)$ Find the solution set of the inequality $ax^{2}-5x+a^{2}-1 > 0$.

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