
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 $f(x)=ax^{2}+(b-1)x+3$ $(a\neq 0)$, <br/>$(1)$ If the solution set of the inequality $f(x) \gt 0$ is $(-1,3)$, find the value of $2a+b$; <br/>$(2)$ If $f(1)=5$, $b \gt -1$, find the minimum value of $\frac{1}{|a|}+\frac{4|a|}{b+1}$; <br/>$(3)$ If $b=-a-3$, find the solution set of the inequality $f(x) \lt -2x+1$.

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