
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 $n \geqslant 2$ be an integer. Alice and Bob play a game involving a country consisting of $n$ islands. Exactly two of these $n$ islands have a factory. Initially, there are no bridges in the country. Alice and Bob take turns playing as follows. On 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 $I_{2}$ has a factory or is connected to a factory by a series of bridges (in fact, access to a factory is necessary for construction).

As soon as a player builds a bridge that allows passage from one factory to the other, that player loses the game (this triggers an industrial war between the two factories). If Alice goes first, determine for which $n \geqslant 2$ she has a winning strategy.

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