```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")
    for drift in drifts:
        if not isinstance(drift, int):
            raise ValueError("each drift must be an integer")

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

    for drift in drifts:
        new_temp = temp + power if heater_on else temp
        new_temp += drift
        if heater_on and new_temp < low or not heater_on and new_temp >= high:
            heater_on = not heater_on
            switches += 1
        if new_temp < coldest:
            coldest = new_temp
        temp = new_temp

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