First step: things to look for to start solving the puzzle
Islands: using the Pigeon Hole Theorem, we can determine some line segments (note that the outcome listed here only guarantees certain lines but IS NOT final)
Doing this process repeatedly throughout the puzzle should be enough to solve it.
Each connected bridge benefits tremendously because it rules out many possibilities. For example:
Finally, two unconnected sets of islands should not be created, as in the "solution" below:
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