
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 \( n \geq 2 \) be an integer. Alice and Bob play a game concerning a country made of \( n \) islands. Exactly two of those \( n \) islands have a factory. Initially, there is no bridge in the country. Alice and Bob take turns in the following way. In each turn, the player must build a bridge between two different islands \( I_{1} \) and \( I_{2} \) such that:

- \( I_{1} \) and \( I_{2} \) are not already connected by a bridge;
- at least one of the two islands \( I_{1} \) and \( I_{2} \) is connected by a series of bridges to an island with a factory (or has a factory itself).

As soon as a player builds a bridge that makes it possible to go from one factory to the other, this player loses the game. Determine, for each \( n \geq 2 \), who has a winning strategy if Alice starts.

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