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Both codes P0670 and P0683 could be set by a couple of things,
Wire harnessGlow plug control module
At the glow plug control module, check for battery voltage on the Red wire at terminal #9 of the Black (BK) connector from the PCM power relay circuit.Check for a ground on the Pink/Orange (PK/OG) wire at terminal #8 of the Green (GN) connector when the Wait to Start light is on. If there is no ground on the PK/OG wire, recheck at the Powertrain Control Module (PCM) pin 101 Violet/Orange (VT/OG). If the VT/OG wire is grounded when the key is cycled on and the Wait to Start light is on, locate and repair the open circuit in between the PCM and the glow plug control module. Start with checking for problems at connector C110 (42 terminal connector near the LH valve cover). Repair the circuit as needed and retest.