
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
2. In each part of this problem, cups are arranged in a circle and numbered $1,2,3$, and so on. A ball is placed in cup 1. Then, moving clockwise around the circle, a ball is placed in every $n$th cup. The process ends when cup 1 contains two balls.

(a) There are 12 cups in the circle and a ball is placed in every 5th cup, beginning and ending with cup 1. List, in order, the cups in which the balls are placed.

(b) There are 9 cups in the circle and a ball is placed in every 6th cup, beginning and ending with cup 1. List the numbers of the cups that do not receive a ball.

(c) There are 120 cups in the circle and a ball is placed in every 3rd cup, beginning and ending with cup 1. How many cups do not contain at least one ball when the process is complete? Explain how you obtained your answer.

(d) There are 1000 cups in the circle and a ball is placed in every 7th cup, beginning and ending with cup 1. Determine the number of the cup into which the 338th ball is placed.

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