
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
After playing basketball, A, B, and C play a game of "Rock, Paper, Scissors" to decide who will treat the others to water. The rules are as follows: Rock beats Scissors, Scissors beat Paper, and Paper beats Rock. Each round of the game only involves two people. A and B play first, then the loser plays against C. The final loser pays for the drinks, and the choice of Rock, Paper, or Scissors by each player is random.

1. Calculate the probability that A wins against B with no more than two throws.
2. Calculate the probability that after two throws among the three players, it is determined that C will treat.
3. Calculate the probability that over the course of three days, exactly two days end with C being determined as the one who will treat after two throws (with each day's game being independent).

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