```python
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")
    
    for d in drifts:
        if isinstance(d, bool) or not isinstance(d, int):
            raise ValueError("drifts must contain integers")

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

    for d in drifts:
        # Read temperature first: switch heater based on current temp
        if temp < low:
            if not heater_on:
                heater_on = True
                switches += 1
        elif temp >= high:
            if heater_on:
                heater_on = False
                switches += 1
        
        # Temperature moves by drift + power if heater is on
        temp += d
        if heater_on:
            temp += power
            heated += 1
        
        # Update coldest
        if temp < coldest:
            coldest = temp

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