
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
The Bank of Zürich issues coins with an  $H$  on one side and a  $T$  on the other side. Alice has  $n$  of these coins arranged in a line from left to right. She repeatedly performs the following operation: if some coin is showing its  $H$  side, Alice chooses a group of consecutive coins (this group must contain at least one coin) and flips all of them; otherwise, all coins show  $T$  and Alice stops. For instance, if  $n = 3$ , Alice may perform the following operations:  $THT \to HTH \to HHH \to TTH \to TTT$ . She might also choose to perform the operation  $THT \to TTT$ .

For each initial configuration  $C$ , let  $m(C)$  be the minimal number of operations that Alice must perform. For example,  $m(THT) = 1$  and  $m(TTT) = 0$ . For every integer  $n \geq 1$ , determine the largest value of  $m(C)$  over all  $2^n$  possible initial configurations  $C$ .

*Massimiliano Foschi, Italy*

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