French-American War in 1797

Say that the US does not approach France's seizure of American ships diplomatically and declares war. What would happen and who would win?
 
I started (and didn't finish) a timeline on this earlier this year.

The US and Great Britain are brought closer together.

The nascent American Navy (aided by Britain), win a few victories.

Adams is reelected.

When Napoleon filches Louisiana back from Spain, the US invades it rather than buys.

Britain quietly agrees to end the impressments against their pseudo-ally.

No War of 1812.
 
Which ironically is probably the best way to have Canada annexed by the U.S. With no War of 1812 and better Anglo-American relations Americans will continue to move into Canada and Britain will do less to promote a separate Canadian identity.

Benjamin
 
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