
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 \(a\) and \(b\) be constants. Suppose that the equation \[\frac{(x+a)(x+b)(x+10)}{(x+4)^2} = 0\]has exactly \(3\) distinct roots, while the equation \[\frac{(x+2a)(x+4)(x+7)}{(x+b)(x+10)} = 0\]has exactly \(1\) distinct root. Compute \(100a + b.\)

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