
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 $c$ and $d$ be constants. Suppose that the equation \[\frac{(x+c)(x+d)(x+10)}{(x+2)^2} = 0\] has exactly $3$ distinct roots, while the equation \[\frac{(x+2c)(x+4)(x+8)}{(x+d)(x+10)} = 0\] has exactly $1$ distinct root. Compute $200c + d.$

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