
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
Let $A$ be a subset of $\{0, 1, 2, \ldots, 29\}$ such that for any integer $k$ and any numbers $a, b$ in $A$ (where $a$ and $b$ can be the same), $a + b + 30k$ is not the product of two consecutive integers. Determine the subset $A$ with the maximum number of elements possible.

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