
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
Draw a line segment of length 1, and denote it by $C_{1}$. Remove the central third (without removing the endpoints). Denote by $C_{2}$ what remains. Now, remove the central third (without the endpoints) from each line segment in $C_{2}$. Denote by $C_{3}$ what remains. We can continue this process, at each stage removing the central third of each segment in $C_{n}$ to form $C_{n+1}$.

(a) Draw $C_{1}, C_{2}$, and $C_{3}$, indicating the numbers at the endpoints of the segments.

(b) Which of the following points belong to the Cantor set? $\frac{1}{3}, \frac{4}{9}, \frac{3}{81}, \frac{4}{81}$.

(c) What are the lengths of $C_{3}, C_{4}$, and $C_{5}$? Can you find an expression for the length of $C_{n}$?

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