Teknocracy5,
I'm not certain how your 1998 V6 Mystique w/ATX is set-up vs. my 1995 V6 Contour w/MTX but here is how I did it on my car:
Jack-up the front end of the car and support it with jack stands (Do Not Use Ramps). Remove the splash panel that is attached under the front end of the car just behind the fascia; mine was attached with 9 Torx-head bolts. There should be a power steering fluid cooler mounted in front of the AC condenser and engine radiator. Determine which feed line going into the power steering cooler is the one where oil flows out under pressure. This is done by looking at the power steering pump. There will be two hoses on the power steering pump. One of the hoses will look like a heavy duty AC hose and the other will look more like a small heater hose. The high pressure output line is the one that looks like a heavy duty AC hose. The pump forces fluid out through the high pressure output line. The high pressure output line goes to the rack-and-pinion assembly. The fluid then flows out of the rack-and-pinion assembly to the power steering cooler and back to the pump through the "heater hose" line.
Once you've determined which line the fluid flows out of when the engine is running, you will need a decent sized polyethylene or polypropylene bottle (> 1-gallon if possible). You may need some spare fuel line of identical i.d. as the power steering fluid hoses connected to the cooler and a short piece of steel brake line to connect the hose on the car to the fuel line (makes it easier to route old fluid into the bottle). The power steering hoses are probably secured to the cooler with spring-loaded hose clamps so you will need a pair of pliers to remove them.
Put a pan underneath the car where the cooler is located to catch oil drips. Loosen the cap on the power steering reservoir. Disconnect the power steering hose I keep referring to and route it into the bottle. A good amount of fluid should drain out due to gravity with the engine off. Start up the car and turn the steering wheel fully left-to-right and back several times until it appears that no more fluid is being ejected into the bottle. Shut the engine off. Reconnect the power steering hose to the cooler. Fill the reservoir with fresh power steering fluid. Start the car and add fluid as necessary while turning the steering wheel completely left-to-right and back. Once the system appears full and purged of air, turn the engine off.
Repeat the draining, pressure purging, refilling & air purging three more times. Fill a fourth time but you should not have to drain it again. Put the car back together and you should be all set.
Of course, inspect the power steering fluid to see if it has cleaned-up in the reservoir. I don't know if you have a heavy contamination that has produced solid deposits that will slowly redissolve into the new fluid. If this is the case, it could take several purge cycles to clean the crap out. It may be necessary to drive the car around a bit to get the fluid warm to help flush-out all the garbage.
I had a 5-gallon bottle and didn't need any assistance in holding the bottle in place when forcing fluid out under pressure. The pressure output was not dangerously high but you may need somebody to hold the bottle so it doesn't get thrown around. Put a rag around the bottle opening to contain any fluid spray from escaping. I used ~ 3/4 gallon of power steering fluid to do this purge on my Contour.
If anybody can step-up and say "Hey, the 1998 Mystique isn't layed-out the same way as I described above", please help out.
Dave Keller
1995 Contour SE, V6, MTX