
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
In a chemistry experiment, a tube contains 100 particles, 68 on the right and 32 on the left. Each second, if there are  $a$  particles on the left side of the tube, some number  $n$  of these particles move to the right side, where  $n \in \{0,1,\dots,a\}$  is chosen uniformly at random. In a similar manner, some number of the particles from the right side of the tube move to the left, at the same time. The experiment ends at the moment when all particles are on the same side of the tube. The probability that all particles end on the left side is  $\tfrac{a}{b}$  for relatively prime positive integers  $a$  and  $b$ .  Compute  $100a+b$ .

*Proposed by Alvin Zou*

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