
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
Let $y = ax^{2} + x - b$ where $a \in \mathbb{R}$ and $b \in \mathbb{R}$.  
$(1)$ Solve the inequality with respect to $x$: $y < (a-1)x^{2} + (b+2)x - 2b$;  
$(2)$ When $a > 0$ and $b > 1$, let the solution set of the inequality $y > 0$ be $P$, and the set $Q = \{x | -2-t < x < -2+t\}$. If for any positive number $t$, $P \cap Q \neq \varnothing$, find the maximum value of $\frac{1}{a} - \frac{1}{b}$.

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