I'm certainly no expert tech, but I believe the cold start valve is energized by one of the pins on the fuel pump relay. Some of those pins are only hot when an RPM (cranking) signal is sent by the ignition, but maybe someone can enlighten us on that. You might check for current next time you get it started. The coolant sensors I've tested are measured by resistance at different temperatures, across the pins on the sensor. If it's four pins, measure across diagonal pins. The resisitance should range from about 5600 ohms at 0 degrees C to approx 600 ohms at 80 C. You should probably get a service manual and double check me for your model.

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1983 380SL
1995 C280
1995 S500 coupe
1990 Mazda Miata
2004 Suzuki Hayabusa