Need help finding a part number

[bmwlover0725]

I need a part number to the hose that goes from the rear of the intake into the firewall. It looks like it may be a brake booster one, but I can't find it in the brake booster diagram. The connector pieces on either side broke when I removed my manifold last time and it's being held up there by suction and pressure from whats sitting directly under it. I'd like to fix this the right way. I'm obviously not going to route a hose through the firewall, but if I can find the hose, then I can at least take the connector from it and swap it onto my hose that's already in the car just by removing the hose clamp around it.

tl;dr - I need the part number for the larger hose at the rear of the intake and I also would like to know what its function is.

EDIT: I found it here - https://www.realoem.com/bmw/enUS/sho...diagId=13_1241

It's part of the idle actuator. Part number 6. And it costs over 100 bucks everywhere. What an absolute ripoff. Oh well... nothing I can do but replace it I suppose.