\(f(n)\) gives the sum of the cubed digits of some positive integer \(n.\) For example, \(f(123)=1^3+2^3+3^3=36.\)

If we repeatedly apply this process on each previous result, the following two different behaviors may arise:

- Arrive at a
**fixed point**: For example, beginning with 3, we eventually arrive at 153, and \(f(153)=153.\) We say that 153 is a fixed point.

- Arrive at a
**limit cycle**: For example, beginning with 4, we eventually arrive at 133, and \(f(133) = 55 \implies f(55) = 250 \implies f(250)=133.\) We say that \(\{55,133,250\}\) is a limit cycle.

The fixed points and limit cycles of this process all belong to the **limit set**.

Find the **sum of all the numbers** in the limit set (including the four in pink found above).

**Note**: A coding environment is provided below:

**Bonus:** Prove that the limit set actually contains finitely many numbers.

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