```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")
    
    current_temp = start
    heater_on = False
    heated_ticks = 0
    switches = 0
    coldest = start
    
    for drift in drifts:
        if isinstance(drift, bool) or not isinstance(drift, int):
            raise ValueError("drift must be an integer")
        
        if heater_on:
            current_temp += power
        
        current_temp += drift
        coldest = min(coldest, current_temp)
        
        if current_temp < low and not heater_on:
            heater_on = True
            switches += 1
        elif current_temp >= high and heater_on:
            heater_on = False
            switches += 1
        
        if heater_on:
            heated_ticks += 1
    
    return {"temp": current_temp, "heated": heated_ticks, "switches": switches, "coldest": coldest}
```