
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
Suppose we have  $10$  balls and  $10$  colors. For each ball, we (independently) color it one of the  $10$  colors, then group the balls together by color at the end. If  $S$  is the expected value of the square of the number of distinct colors used on the balls, find the sum of the digits of  $S$  written as a decimal.

*Proposed by Michael Kural*

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