
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 
\[\frac{(x+p)(x+q)(x-15)}{(x-5)^2} = 0\]
has exactly $3$ distinct roots, while the equation 
\[\frac{(x-2p)(x-5)(x+10)}{(x+q)(x-15)} = 0\]
has exactly $2$ distinct roots. Compute $100p + q.$

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