
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
Two circles \(\Gamma_{1}\) and \(\Gamma_{2}\) of radius 1 and 2, respectively, are centered at the origin. A particle is placed at \((2,0)\) and is shot towards \(\Gamma_{1}\). When it reaches \(\Gamma_{1}\), it bounces off the circumference and heads back towards \(\Gamma_{2}\). The particle continues bouncing off the two circles in this fashion.

If the particle is shot at an acute angle \(\theta\) above the \(x\)-axis, it will bounce 11 times before returning to \((2,0)\) for the first time. If \(\cot \theta = a - \sqrt{b}\) for positive integers \(a\) and \(b\), compute \(100a + b\).

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