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The HMS Prince is probably the easiest of all the legendary ship encounters. However whilst you battle this ship, you will also have to deal with a thick fog that can make visibility a little tricky at times. This legendary ship has a propensity to chain spam its mortar attacks and hit you with its massively damaging broadside attacks as well. Fortunately, if you can avoid these attacks the difficulty of the encounter drops significantly.
The easiest way to defeat this ship is to try and take the mortars out of the equation. That means trying to stay as close as you can to the rear of the enemy vessel, either by constantly maintaining a position slightly off to the side or following directly behind it. From here you should still be able to use most of your attacks but the major focus of your offensive efforts should be on dropping mortars and using the chain shot.
When the ship turns, which it will, use the opportunity to hit it with a few heavy shots whilst you try to re-position yourself behind it again.