venting a 400 watt cooltube in a closet

okay so i'm going to be putting the laides into flowering in a few weeks here and i recently just placed my order for a 400 watt hps with lumetek ballast with a 6" cooltube. My question is, will a 4" inline fan be enough to cool down the light (with duct reducers) as well as feed a carbon scrubber that will be inline with my exhaust? Or will it be necessary to upgrade to a 6"? My grow space is 4'x3'7'. any imput would be appreciated. (i'm also on a budget so lets keep that in consideration haha.)

I use a 6in vortex fan as my outtake and a 6in inline as intake in about a 2'Wx3'Lx8'H bathtub using a 400whps on a winged hood. Temps stay around 65ish when the lights are on. If i were u i would get maybe 2 inline fans at 6in or atleast a higher powered fan and a small inline fan.

maybe a 6" inline fan for exhaust (and scrubber as well as venting for cooltube) and maybe 2 6" in-line duct booster fans for my intake into room?

cant say if it will work or not, best way to find out is to set it up & see
if it does't work & you are on a budget, 2 pc fans can work. 1 on intake & 1 on the exhaust w/out a carbon filter. just take fresh air from outside your grow and run it through the light & out the room. since the air never comes in contact w/ the plants, it wont pick up the smell so you dont need a filter. also keep ducts straight & short. I did this a few yrs ago w/ 3 feet of ducting and a sweeping 90 bend feeding to the light and about 6 feet straight out(all 4 in) on a 400 watt mh running 18 hr days without a problem. temps in the closet only went up about 3 degrees by the end of the day cycle. air was from basement so it was fairly cool (under 70).

the inline fans are not very powerful. if you can seal your area well enough having one for exhaust and two for intake my work.