
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 into the forest and collectively brought back 450 mushrooms (each one brought home at least one mushroom). A boy named Petya, upon learning this, declared: "There must be at least two of them who brought back the same number of mushrooms!" What is the smallest $n$ for which Petya is certainly correct? Don't forget to justify your answer.

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