Help: 1/1/1 lpc

I cannot figure out the instructions for my pattern - I have done basic cables before, but for some reason I can’t figure out what I’m supposed to do here. This is a portion of Counter Turn hat pattern (part of the Salchow Trio pattern).

the pattner says:

1/1/1 LPC - Slip 1 st on cn and hold in front, k the 2nd st from the tip of the LN through the front, do not drop the worked stitch just yet, p the 1st st on LN, drop both worked sts from LN, kn1 from cn

When is says p the 1st st on LN, do they mean the “new” 1st st, but that would be increasing a stitch, right?

Yes, it’s the [I]new[/I] first and the new second stitch. Instructions for sts on the left hand needle are always referring to the current sts on the needle.
The purl into the first stitch won’t be an increase because you’ll then drop the old first stitch (and the second stitch) off the needle. So you knit into the second stitch but leave the stitch on the needle, purl into the first stitch and slip both old sts off the left hand needle.
You can also do it with 2 cable needles but it’s a bit cumbersome. Slip 1 st to cn and hold in front, slip next st to second cn and hold in back, k1, p1 from back cn, k1 from front cn.