
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
a) There are $2n+1$ batteries ($n > 2$). It is known that there are one more good batteries than bad ones among them, but it is unknown which specific batteries are good or bad. Two batteries are inserted into a flashlight at a time, and the flashlight only lights up if both of them are good. What is the minimum number of such attempts needed to guarantee that the flashlight lights up?

b) The same problem, but with $2n$ batteries ($n > 2$), where there are equal numbers of good and bad batteries.

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