$f(n)$ 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.

Let the **limit set** be the set of all fixed points and limit cycles in the range of $f(n).$

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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