
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
Suppose  $A\subseteq \{0,1,\dots,29\}$ . It satisfies that for any integer  $k$  and any two members  $a,b\in A$ ( $a,b$  is allowed to be same),  $a+b+30k$  is always not the product of two consecutive integers. Please find  $A$  with largest possible cardinality.

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