
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
For a real number \( r \), the quadratics \( x^{2}+(r-1) x+6 \) and \( x^{2}+(2 r+1) x+22 \) have a common real root. The sum of the possible values of \( r \) can be expressed as \(\frac{a}{b}\), where \( a \) and \( b \) are relatively prime positive integers. Compute \( 100a + b \).

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