
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 $y = ax^{2} + (b-2)x + 3$.  
$(1)$ If the solution set of the inequality $y > 0$ with respect to $x$ is $\{x | -1 < x < 3\}$, find the solution set of $y \geq 4$.  
$(2)$ If $x = 1$ and $y = 2$, $a > 0$, $b > 0$, find the minimum value of $\frac{1}{a} + \frac{4}{b}$.

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