```python
def fold_machine(machine: dict) -> dict:
    alphabet = machine.get("alphabet")
    states = machine.get("states")
    start = machine.get("start")
    accepting = machine.get("accepting")
    moves = machine.get("moves")

    if not alphabet or not states or not start or not accepting or not moves:
        raise ValueError("Invalid input machine.")

    if len(alphabet) != len(set(alphabet)) or any(not isinstance(s, str) or s == "" for s in alphabet):
        raise ValueError("Alphabet must be a list of distinct non-empty strings.")
    if len(states) != len(set(states)) or any(not isinstance(s, str) or s == "" for s in states):
        raise ValueError("States must be a list of distinct non-empty strings.")
    if start not in states:
        raise ValueError(f"Start '{start}' is not a listed state.")
    if len(accepting) != len(set(accepting)) or any(s not in states for s in accepting):
        raise ValueError("Accepting states must be a list of distinct names drawn from the states.")
    if any(state not in states or symbol not in alphabet for state, symbol, _ in moves):
        raise ValueError("Moves must name existing states and symbols.")

    reachable = set()
    queue = [start]
    while queue:
        current_state = queue.pop(0)
        if current_state not in reachable:
            reachable.add(current_state)
            for _, symbol, next_state in moves:
                if symbol == "" or next_state == "":
                    raise ValueError("Symbol and state names must be non-empty strings.")
                if (current_state, symbol) in [(s, sy) for s, sy, _ in moves]:
                    queue.append(next_state)

    unreachable_states = states - reachable
    machine["moves"] = [move for move in moves if move[0] not in unreachable_states and move[2] not in unreachable_states]

    equivalent_classes = []
    while len(states) > 0:
        current_state = states.pop(0)
        class_ = {current_state}
        queue = [(current_state, symbol) for symbol in alphabet]
        while queue:
            current_state, symbol = queue.pop(0)
            next_state = [move[2] for move in moves if move[0] == current_state and move[1] == symbol][0]
            if next_state not in class_:
                class_.add(next_state)
                if next_state in states:
                    states.remove(next_state)
                    queue.extend([(next_state, sy) for sy in alphabet])
        equivalent_classes.append(class_)

    machine["moves"] = [
        [equivalent_classes.index(group), symbol, equivalent_classes.index([move[2] for move in moves if move[0] == state and move[1] == symbol][0])]
        for group in equivalent_classes
        for state in group
        for symbol in alphabet
    ]

    groups = {i: sorted(class_) for i, class_ in enumerate(equivalent_classes)}
    machine["size"] = len(groups)
    machine["start"] = 0
    machine["accepting"] = [groups.index(sorted([s for s in states if s in accepting])) for s in states if s in accepting]
    machine["moves"] = sorted(
        machine["moves"],
        key=lambda x: (x[0], alphabet.index(x[1])),
    )

    return machine
```