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If you still have your Integritys on the car, forget it.
I switched to 235 Michelin Cross-Terrains, and have fought every kind of turf imaginable with 99 AWD here in Mexico. The thing you have to watch for is the shocks, and not take to deep a ride. Moreover, your air filter can get clogged very quickly with dust, and set off the OBD II Sensor.
I have seen Pathfinders (no QX4s sold in Mexico) get stuck in stuff down here, that my RX did not.