How did The Police discover reggae?

So if The Police were a punk band originally, where did the idea to play reggae come from?

Drummer Stewart Copeland explains in his memoir Strange Things Happen:

“At the punk clubs, they would play punk music.  But you can’t play punk all night without giving everybody headaches, even punks.  So they had to have some variety in the energy levels.  But there’s no such thing as mellow punk.  So they would alternate the punk music with reggae, particularly dub reggae.  

“This got all the punk bands thinking, ‘Damn, we should try to play that stuff.’  But none of them were very good at it.  The Clash, The Damned, The Sex Pistols—all those groups had great charisma and great enthusiasm, but The Police just were better at it than anyone else!  We were better at just about everything than anyone else because we were real musicians.”

Andy Summers realizes his dream

In his memoir, One Train Later, Andy Summers described Sting’s demo for "Every Breath You Take" as a “Hammond organ thing that sounded like Billy Preston.”  But Sting told him to make the song his own.  

“It’s a simple chord progression and shouldn’t prove a problem,” he writes.  “The track rolls and I play a sequence of intervals that makes it sound like The Police.  It is clean, succinct, immediately identifiable.  I play it straight through in one take.  There is a brief silence, and then everyone in the control room stands up and cheers.

“With this lick, I realize a dream that I have cherished since first picking up the guitar as a teenager—to at least once in my life make something that would go around the world, create a lick that guitarists everywhere would play.”