
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
$n$ mushroom pickers went to the forest and collectively brought back 162 mushrooms (each brought home at least one mushroom). A boy named Petya, upon hearing this, claimed: "Some two of them must have brought the same number of mushrooms!" What is the smallest $n$ for which Petya is definitely correct? Don't forget to justify your answer.

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