
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 $f(x)=ax^{2}+4x+3$.  
$(1)$ If the solution set of the inequality $f(x) \gt 0$ is $(b,1)$, find the values of $a$ and $b$.  
$(2)$ If $a \gt 0$, find the solution set of the inequality $f(x) \gt -ax-1$.

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