
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
Let  $n$  be an even positive integer. Alice and Bob play the following game. Before the start of the game, Alice chooses a set  $S$  containing  $m$  integers and announces it to Bob. The players then alternate turns, with Bob going first, choosing  $i\in\{1,2,\dots, n\}$  that has not been chosen and setting the value of  $v_i$  to either  $0$  or  $1$ . At the end of the game, when all of  $v_1,v_2,\dots,v_n$  have been set, the expression   $$ E=v_1\cdot 2^0  + v_2 \cdot 2^1 + \dots + v_n \cdot 2^{n-1} $$   is calculated. Determine the minimum  $m$  such that Alice can always ensure that  $E\in S$  regardless of how Bob plays.

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