
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 gatherers went to the forest and brought back a total of 338 mushrooms (it is possible that some of the gatherers did not bring back any mushrooms). A boy named Petya, upon learning this, stated, "Some two of them must have brought an identical number of mushrooms!" For what smallest $n$ will Petya certainly be correct? Provide a justification for your answer.

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