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Grey smoke on hard acceleration makes it sound like your car is running too rich. Have you worked out
your fuel consumption yet?
It could perhaps be a worn oxygen sensor since the car is 13 years old, despite only 78k.
You need to get a paperclip and connect it to the diagnostics port on the engine (it has diagnostics written on it) Find out what error codes (if any) are appearing. Then you need to find the troubleshooting guide of the LS400 manual (someone on here might have it), to tell you what each error code means. If you lift up the diagnostics cover, it should say underneath which terminals are which.
Don't worry, if you do need something fixing, people on here can help you get it fixed at a reasonable price.
Ok see this link, and also read this too.
http://www.off-road.com/toyota/tech/codes/There is a way to retrieve the trouble code stored in the computer. I only have trouble codes for the A245E, but if it's the same problem, then the code will be the same for the A340E. All you need to retrieve the code is a jumper wire to bridge terminals in the diagnostic port (in the engine compartment on your OBD-I car). This will NOT work on OBD-II equipped cars ('95+ LS400s are OBD-II).
The procedure is:
1. Turn the ignition switch to ON (do not start the engine)
2. Bridge the terminals E1 and TE1 of the data link connector with a paperclip or jumper wire.
3. The code stored is indicated by the number of flashes of the "O/D OFF" light. The light will blink the number of the first digit of the code, then pause, then blink the number of the second digit of the code, then a longer pause, then will either move on to another code if a second one has been stored, or will flash the first one again.