
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
The sum of two numbers is 50, and their product is 25. If the sum of their reciprocals is $a$, find $a$.

If the lines $ax + 2y + 1 = 0$ and $3x + by + 5 = 0$ are perpendicular, find $b$.

The area of an equilateral triangle is $100\sqrt{3} \mathrm{~cm}^2$. If its perimeter is $p \mathrm{~cm}$, find $p$.

If $x^3 - 2x^2 + px + q$ is divisible by $x + 2$, find $q$.

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