```uclid
module main {
    var x : int;
    type action_t = enum {IDLE, ACTION_A, ACTION_B};
    var action : action_t;

    init {
        x = 3;
        action = IDLE; // It's important to initialize 'action' in init block
    }

    next {
        case
        is_nondeterministic_choice: 
            action' = nondet(action_t); // Corrected nondeterministic assignment to action
        else:
            case
            action == ACTION_A: x' = 2 * x;
            action == ACTION_B: x' = x + 1;
            action == IDLE: x' = x;
            esac;
        esac;
    }

    control {
        while (true) {
            if (nondet()) {
                havoc action;
            } 
        }
    }
}
```
In this corrected version of the code:
1. Rewrote the `next` block with a single block encapsulating action updates and state variable updates.
2. Replaced syntactically incorrect `next` function with a case statement to handle the nondeterministic assignment.
3. Modified the control loop to properly utilize nondeterminism (`nondet`) and state modification (`havoc`), assuming changes apply based on model checker/framework behavior and their handling of nondeterministic choices.