
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 positive integer  $n (n \geq 2)$ , find the largest positive integer  $\lambda$  satisfying :
For  $n$  bags, if every bag contains some balls whose weights are all integer powers of  $2$  (the weights of balls in a bag may not be distinct), and the total weights of balls in every bag are equal, then there exists a weight among these balls such that the total number of balls with this weight is at least  $\lambda$ .

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