
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 in terms of $x$, $ax^2+2x+b>0$, is $\{x|x\neq -\frac{1}{a}\}$, find the minimum value of $\frac{a^2+b^2+7}{a-b}$ (where $a>b$).

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