Cleaning tap lines and taps with one pass of solution

I am looking for some advice on how to clean all my tap lines without a bunch of disassembly.

My thought is to daisy chain all the lines but I don’t know how to attach the keg line out couplers together. I want to run a pump solution out of a 5 gallon pale of and into one of the 4 keg out couplers then through open tap #1. Then I want it to go from open tap #1 (with a larger id line that will slip over the Tap opening) to tap #2. The solution would go through tap #2 backwards to the next keg coupler. From this point I want to couple two of the keg out couplers together then the solution will go out tap #3 and in backwards through tap #4 where it would then go through the last coupler back into the pale. I am not sure how to do the plumbing? How are you all doing this task of cleaning the lines and taps? Is there a better way?

IMO, you're making it far more complicated than it should be.

My method is as follows:
Once a keg kicks, I simply remove it from the brew fridge, release the pressure inside it, rinse it out with hot water and then mix up about a gallon of hot PBW in it. I shake it really well to get it everywhere inside the keg. After giving it a decent soak time (depends on how the keg looks inside) I connect it to a gas feed and then connect the liquid QD that it was using back onto it. I run at least a half gallon of PBW solution through the hose and tap/faucet assembly (into a quart measuring cup, or gallon pail). I discard the 'dirty' solution from both the tap and what's left in the keg. I then rinse out the keg a couple of times (to get the PBW out of it) and fill it [again] with at least warm water (about a gallon or two, no measuring is done this time). I run at least a couple of quarts through the hose setup again. Then I put a few quarts of StarSan solution into the keg. Again, I shake the F out of it to fully coat the inside of the keg. Remember, each time I fill the keg with a different liquid, I make sure to seal it, then it gets it's own gas feed to push through the hose. I run at least a quart or two of StarSan through the setup again. If I don't have a keg to take the spot of the one that kicked, then I leave the line with StarSan in it. Otherwise, I connect it up and run it until the StarSan is pushed out (I discard that small amount) and beer is flowing freely.

I've used this method with success over the past several months since I started kegging. I have three kegs on tap at all times, with a fourth carbonating in the fridge. It doesn't take that long to run through the steps, and I feel it's well worth the time spent. I'd rather spend a bit of time to ensure things are 100% than think I'm being clever only to set myself up for bad things later.

Oh, and I do take apart the kegs (especially the liquid side) from time to time, or after a keg has rotated through at least a couple of batches to ensure nothing is gumming up the works there.

Perfect, I clean the keg anyway after a batch so this is the way to go, thanks. I tend to overthink things...