[THOUGHT] To solve this problem, we need to count the number of closed-loop sequences of length ( n ) where any two elements are distinct if they are adjacent or separated by exactly one position. This is similar to counting Hamiltonian cycles in a graph with specific constraints.

1. Graph Representation: Represent the problem as a graph where each node is an element from ( V ), and edges connect nodes that can be adjacent or separated by one position in the sequence. The sequence must form a cycle where no two adjacent or one-apart nodes are the same.

2. Constraints: The sequence must satisfy:

   • ( v_i \neq v_{i+1} ) for all ( i ) (adjacent elements are distinct).
   • ( v_i \neq v_{i+2} ) for all ( i ) (elements one position apart are distinct).
   • ( v_n = v_0 ) (closed loop).
   • ( v_1 \neq v_{n-1} ) (boundary condition).

3. Approach: This is a counting problem with constraints. We can model it using dynamic programming or recursion, but for ( n = 10 ), we might need an efficient method due to the size.

4. Sympy or Backtracking: Use symbolic computation or backtracking to count valid sequences. Given the constraints, we can enumerate all possible sequences and count those that meet the conditions.

[/THOUGHT] [PYTHON] from itertools import permutations

def is_valid_sequence(sequence): n = len(sequence) for i in range(n): if sequence[i] == sequence[(i+1)%n]: return False if sequence[i] == sequence[(i+2)%n]: return False if sequence[1] == sequence[-2]: return False return True

V = set(range(1, 12)) n = 10

We need to count sequences of length n where v_n = v_0 and the above conditions hold.

This is equivalent to counting cycles of length n-1 with the constraints.

Generate all possible permutations of length n-1 with distinct elements, then check the constraints.

count = 0 for perm in permutations(V, n-1): seq = list(perm) + [perm[0]] # Close the loop if is_valid_sequence(seq): count += 1

Each cycle is counted (n-1)! times due to permutations, but we need distinct cycles.

However, since the sequence is fixed in order, we don't divide here.

print(count) [/PYTHON]

[THOUGHT] The Python output shows that no valid sequences of length (10) satisfy the given constraints. This is the exact result based on the computation. [/THOUGHT] \boxed{0}