```python
def run_thermostat(start: int, low: int, high: int, power: int, drifts: list) -> dict:
    if not all(isinstance(x, int) for x in [start, low, high, power]):
        raise ValueError("Start, low, high, and power must be integers.")
    if low >= high:
        raise ValueError("Low must lie strictly below high.")
    if power <= 0:
        raise ValueError("Power must be a positive integer.")
    if not isinstance(drifts, list) or not all(isinstance(x, int) for x in drifts):
        raise ValueError("Drifts must be a list of integers.")

    temp = start
    heated = 0
    switches = 0
    coldest = temp

    heater_on = False

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

        if temp < low and not heater_on:
            heater_on = True
            switches += 1
            coldest = min(coldest, temp)
        elif temp >= high and heater_on:
            heater_on = False
            switches += 1

        coldest = min(coldest, temp)

    return {
        "temp": temp,
        "heated": heated if heater_on else heated + 1,
        "switches": switches,
        "coldest": coldest
    }
```