
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
Adam has a single stack of  $3 \cdot 2^n$  rocks, where  $n$  is a nonnegative integer. Each move, Adam can either split an existing stack into two new stacks whose sizes differ by  $0$  or  $1$ , or he can combine two existing stacks into one new stack. 
    
Adam keeps performing such moves until he eventually gets at least one stack with  $2^n$  rocks. Find, with proof, the minimum possible number of times Adam could have combined two stacks.

*Proposed by Anthony Wang*

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