```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")
    # Walk the drifts, switching the heater by the two-point rule,
    # and track the heated ticks, the switches and the coldest reading.
    heated = False
    coldest = start
    switch_count = 0
    for drift in drifts:
        if not isinstance(drift, int):
            raise ValueError("drifts must be a list of integers")
        if heated:
            if drift > 0:
                start += drift
                heated = False
            else:
                start += drift
                switch_count += 1
        else:
            if drift < 0:
                start += drift
                heated = True
            else:
                start += drift
                switch_count += 1
            coldest = min(coldest, start)
    return {"temp": start, "heated": heated, "switches": switch_count, "coldest": coldest}
```