
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
 $23$  frat brothers are sitting in a circle. One, call him Alex, starts with a gallon of water. On the first turn, Alex gives each person in the circle some rational fraction of his water. On each subsequent turn, every person with water uses the same scheme as Alex did to distribute his water, but in relation to themselves. For instance, suppose Alex gave  $\frac{1}{2}$  and  $\frac{1}{6}$  of his water to his left and right neighbors respectively on the first turn and kept  $\frac{1}{3}$  for himself. On each subsequent turn everyone gives  $\frac{1}{2}$  and  $\frac{1}{6}$  of the water they started the turn with to their left and right neighbors, respectively, and keep the final third for themselves. After  $23$  turns, Alex again has a gallon of water. What possibilities are there for the scheme he used in the first turn? 
(Note: you may find it useful to know that  $1+x+x^2+\cdot +x^{23}$  has no polynomial factors with rational coefficients)

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