```python

def run_thermostat(start: int, low: int, high: int, power: int, drifts: list) -> dict:
    if not isinstance(start, int):
        raise ValueError("start must be an integer")
    if not isinstance(low, int):
        raise ValueError("low must be an integer")
    if not isinstance(high, int):
        raise ValueError("high must be an integer")
    if low >= high:
        raise ValueError("low must be strictly below high")
    if power <= 0 or not isinstance(power, int):
        raise ValueError("power must be a positive integer")
    if not isinstance(drifts, list):
        raise ValueError("drifts must be a list")
    
    temp = start
    heated_ticks = 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
        
        if temp < coldest:
            coldest = temp
        
        if temp < low and not heater_on:
            heater_on = True
            switches += 1
        elif temp >= high and heater_on:
            heater_on = False
            switches += 1
    
    return {"temp": temp, "heated": heated_ticks, "switches": switches, "coldest": coldest}
```