
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 \( A \subseteq \{0, 1, 2, \cdots, 29\} \) such that for any integers \( k \) and any numbers \( a \) and \( b \) (possibly \( a = b \)), the expression \( a + b + 30k \) is not equal to the product of two consecutive integers. Determine the maximum possible number of elements in \( A \).

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