```python
def run_thermostat(start: int, low: int, high: int, power: int, drifts: list) -> dict:
    if not isinstance(start, int) or not isinstance(low, int) or not isinstance(high, int):
        raise ValueError("Start, low, and high must be integers.")
    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 of integers.")
    
    temp = start
    heated = 0
    switches = 0
    coldest = start
    
    for drift in drifts:
        new_temp = temp + drift
        
        if new_temp < low or new_temp > high:
            heated += 1
            switches += 1
            temp = max(low, min(new_temp, high))
        else:
            temp += power
    
        coldest = min(coldest, temp)
    
    return {
        "temp": temp,
        "heated": heated,
        "switches": switches,
        "coldest": coldest
    }
```