
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 that the solution set of the quadratic inequality $ax^{2}+bx+c \lt 0\left(a,b,c\in R\right)$ is $\{x\left|\right.-1 \lt x \lt 3\}$, calculate the minimum value of $b-2c+\frac{1}{a}$.

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