
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 \( A \cong \{0, 1, 2, \cdots, 29\} \) satisfy: for any integer \( k \) and any numbers \( a, b \) in \( A \) ( \( a \) and \( b \) can be the same), \( a + b + 30k \) is not the product of two consecutive integers. Determine all sets \( A \) with the maximum number of elements.

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