
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 $f\left(x\right)=ax^{2}+bx+1$.  
$(1)$ If $f\left(1\right)=2$, and $a \gt 0$, $b \gt 0$, find the minimum value of $\frac{1}{a}+\frac{4}{b}$.  
$(2)$ If $b=-a-1$, solve the inequality $f\left(x\right)\leqslant 0$ with respect to $x$.

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