
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 function $f(x) = x^2 + (3-a)x + 2 + 2a + b$, where $a, b \in \mathbb{R}$:
1. If the solution set of the inequality $f(x) > 0$ is $\{x | x < -4 \text{ or } x > 2\}$, find the values of real numbers $a$ and $b$;
2. If the inequality $f(x) \leq b$ has solutions on the interval $x \in [1, 3]$, find the range of the real number $a$;
3. If the solution set of the inequality $f(x) < 12 + b$ contains exactly three integers, find the range of the real number $a$.

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