
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
A basketball player has a probability of $a$ for scoring 3 points in a single shot, a probability of $b$ for scoring 2 points, and a probability of $c$ ($a$, $b$, $c \in (0, 1)$) for not scoring any points. It is known that the mathematical expectation for the player's points per shot is 2 (not considering other scoring situations). Determine the maximum value of $ab$.

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