Sure, I can help you write a test case generator for the given problem statement. Here's the test case generator in Python:

```python
import random

def generate_precisely_sequence(N, range_min=-100000000, range_max=100000000):
    half_N = N // 2
    first_half = [random.randint(range_min, range_max) for _ in range(half_N)]
    second_half = first_half[::-1]
    return first_half + second_half

def generate_queries(Q, N, x_min=-1000, x_max=1000):
    queries = []
    for _ in range(Q):
        l = random.randint(1, N)
        r = random.randint(l, N)
        x = random.randint(x_min, x_max)
        queries.append((l, r, x))
    return queries

def tcgen():
    # Generate N, which is even and in the range [2, 500,000]
    N = random.randint(1, 250000) * 2

    # Generate sequence S
    S = generate_precisely_sequence(N)

    # Generate Q, which is in the range [1, 100,000]
    Q = random.randint(1, 100000)

    # Generate queries
    queries = generate_queries(Q, N)

    return N, S, Q, queries

def format_test_case(N, S, Q, queries):
    test_case = []
    test_case.append(str(N))
    test_case.append(" ".join(map(str, S)))
    test_case.append(str(Q))
    for query in queries:
        test_case.append(" ".join(map(str, query)))
    return "\n".join(test_case)

if __name__ == "__main__":
    N, S, Q, queries = tcgen()
    test_case = format_test_case(N, S, Q, queries)
    print(test_case)
```

Explanation:

1. **generate_precisely_sequence(N, range_min, range_max)**: This function generates a "precisely" arranged sequence of length \( N \). The sequence is symmetric, so the first half is randomly generated and the second half is the reverse of the first half.

2. **generate_queries(Q, N, x_min, x_max)**: This function generates \( Q \) queries. Each query is a tuple \((l, r, x)\), where \( l \) and \( r \) are indexes within the range of the sequence, and \( x \) is a random integer within the given limits.

3. **tcgen()**: This function coordinates the generation of \( N \), \( S \), \( Q \), and the queries. It uses the previous helper functions to create a consistent test case.

4. **format_test_case(N, S, Q, queries)**: This function formats the generated test case into the required input format for the problem. It returns a string that represents the entire test case.

The `if __name__ == "__main__":` block is for testing and printing the generated test case.

You can run this script to generate a random test case for the given problem statement. Adjust the ranges and limits as necessary to fulfill specific requirements or constraints.