
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
Joe has  $1729$  randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes of sidelengths  $9$  and  $10$  or of sidelengths  $1$  and  $12$  (randomly chosen). These cubes are dipped into white paint. Then two more cubes of sidelengths  $1$  and  $12$  or  $9$  and  $10$  are formed from the same unit cubes, again randomly oriented and randomly arranged, and dipped into paint. Joe continues this process until every side of every unit cube is painted. After how many times of doing this is the expected number of painted faces closest to half of the total?

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