
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
Positive integers (not necessarily unique) are written, one on each face, on two cubes such that when the two cubes are rolled, each integer  $2\leq k\leq12$  appears as the sum of the upper faces with probability  $\frac{6-|7-k|}{36}$ . Compute the greatest possible sum of all the faces on one cube.

*2015 CCA Math Bonanza Tiebreaker Round #3*

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