```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):
        raise ValueError("drifts must be a list")
    if not all(isinstance(d, int) for d in drifts):
        raise ValueError("all drifts must be integers")

    temp = start
    heated = 0
    switches = 0
    coldest = start
    heater_on = False

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

        if temp < coldest:
            coldest = temp

        if temp < low:
            heater_on = True
            switches += 1
        elif temp >= high:
            heater_on = False
            switches += 1
        else:
            heater_on = not heater_on
            switches += 1

        heated += heater_on

    return {"temp": temp, "heated": heated, "switches": switches, "coldest": coldest}
```