
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
Erzsi received the following problem: "Find 5 integers such that when any two of them are multiplied together and then divided by 12, the remainder is always different!" Then solve the same problem, but with 14 instead of 12!

Erzsi realized that it is sufficient to search for the numbers from $1, 2, \ldots, 11$ in the first case, and from $1, 2, \ldots, 13$ in the second case.

Can we help Erzsi further?

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