
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
Mark the Martian and Bark the Bartian live on planet Blok, in the year  $2019$ . Mark and Bark decide to play a game on a  $10 \times 10$  grid of cells. First, Mark randomly generates a subset  $S$  of  $\{1, 2, \dots, 2019\}$  with  $|S|=100$ . Then, Bark writes each of the  $100$  integers in a different cell of the  $10 \times 10$  grid. Afterwards, Bark constructs a solid out of this grid in the following way: for each grid cell, if the number written on it is  $n$ , then she stacks  $n$   $1 \times 1 \times 1$  blocks on top of one other in that cell. Let  $B$  be the largest possible surface area of the resulting solid, including the bottom of the solid, over all possible ways Bark could have inserted the  $100$  integers into the grid of cells. Find the expected value of  $B$  over all possible sets  $S$  Mark could have generated.

*Proposed by Yang Liu*

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