
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
Let \( p \) and \( q \) be constants. Suppose that the equation \[\frac{(x+p)(x+q)(x+5)}{(x+2)^2} = 0\] has exactly \(2\) distinct roots, while the equation \[\frac{(x+2p)(x+2)(x+3)}{(x+q)(x+5)} = 0\] has exactly \(2\) distinct roots. Compute \(100p + q.\)

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