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The PS failure is common on these cars. Since the PS uses a electric pump, any electrical concern could cause this concern.
I would start by checking the PS fluid level. Next disconnect the battery. Hold your foot on the brake pedal for 1 min. Clean and reconnect the cables. With the engine running, place a volt meter across the battery post. Voltage should read between 13-15 volts. A faulty alternator may cause your concern.
Also check all fuses.
Any Mazda dealer will be able to diagnose and repair your concern.
Please let me know if you have any additional questions.
The speedo and ps are both electronic. The do share the same ignition fuses.
Here is a wiring diagram of both systems.
You will need to check all fuses in the engine compartment fuse panel and the on under the glove box.
I would use a test light to probe the top of each fuse through the 2 small holes. With the car running, there should be voltage on both sides off most all fuses.
Please let me know if you need additional diagnosis steps if no blown fuses are found.
#3 in this diagram is the VSS that controls electronic signal to your speedometer. Check connector for damage and remove and inspect plastic gear.