```python
def run_thermostat(start: int, low: int, high: int, power: int, drifts: list) -> dict:
    for bound in (start, low, high):
        if not isinstance(bound, int):
            raise ValueError("temperatures must be integers")
    if low >= high:
        raise ValueError("low must lie strictly below high")
    if not isinstance(power, int) or power <= 0:
        raise ValueError("power must be a positive integer")
    if not isinstance(drifts, list) or not all(isinstance(d, int) for d in drifts):
        raise ValueError("drifts must be a list of integers")

    room_temp = start
    heater_on = False
    heated_ticks = 0
    switches = 0
    coldest_temp = start

    for drift in drifts:
        if heater_on:
            room_temp += power + drift
        else:
            room_temp += drift

        if room_temp < low and not heater_on:
            heater_on = True
            switches += 1
        elif room_temp >= high and heater_on:
            heater_on = False
            switches += 1

        heated_ticks += heater_on
        coldest_temp = min(coldest_temp, room_temp)

    return {"temp": room_temp, "heated": heated_ticks, "switches": switches, "coldest": coldest_temp}
```