[THOUGHT] I will compute the result using Python by calculating the squared length of PQ and the squared magnitude of the cross product of vectors PQ and PR, then multiply them to get the final value. This avoids floating-point operations and ensures integer arithmetic. [/THOUGHT] [PYTHON] import math

P = (2, -1, 3) Q = (-7, 8, -6) R = (4, 0, 1)

Vector PQ = Q - P

PQ = (Q[0] - P[0], Q[1] - P[1], Q[2] - P[2])

Vector PR = R - P

PR = (R[0] - P[0], R[1] - P[1], R[2] - P[2])

Squared length of PQ (ℓ^2)

l_squared = PQ[0]**2 + PQ[1]**2 + PQ[2]**2

Cross product components of PQ and PR

cross_x = PQ[1]*PR[2] - PQ[2]*PR[1] cross_y = PQ[2]*PR[0] - PQ[0]*PR[2] cross_z = PQ[0]*PR[1] - PQ[1]*PR[0]

Squared magnitude of the cross product (||PQ × PR||^2)

cross_magnitude_squared = cross_x2 + cross_y2 + cross_z**2

The expression 4ℓ^2 A^2 equals ℓ^2 * ||PQ × PR||^2

result = l_squared * cross_magnitude_squared print(result) [/PYTHON]

[THOUGHT] The computed result is 511758. This matches the derived value from manual calculation. [/THOUGHT] \boxed{511758}

[THOUGHT] The Python output is 511758, which is the value of (4\ell^2 A^2). [/THOUGHT] \boxed{511758}