
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
Given a positive integer $n (n \geqslant 2)$, find the largest $\lambda$ such that if there are $n$ bags, each bag containing some balls with weights that are integer powers of 2, and the total weight of the balls in each bag is equal, then there must be at least $\lambda$ balls of the same weight. (A bag can contain multiple balls of the same weight.)

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