```
def run_thermostat(start: int, low: int, high: int, power: int, drifts: list) -> dict:
    for bound in (start, low, high):
        if isinstance(bound, bool) or not isinstance(bound, int):
            raise ValueError("temperatures must be integers")
    if low >= high:
        raise ValueError("low must lie strictly below high")
    if isinstance(power, bool) or 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")

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

    for drift in drifts:
        if not isinstance(drift, int):
            raise ValueError("each drift must be an integer")
        
        if heater_on:
            temp += power
        
        temp += drift
        coldest = min(coldest, temp)
        
        if temp < low:
            if not heater_on:
                heater_on = True
                switches += 1
        elif temp >= high:
            if heater_on:
                heater_on = False
                switches += 1
    
    return {"temp": temp, "heated": heated, "switches": switches, "coldest": coldest}
```