[THOUGHT] We need to solve a geometry problem step-by-step.

Given:

• L1: intersection of planes 2x+3y+z=4 and x+2y+z=5.
• L2: passes through P(2,-1,3) and parallel to L1.
• Plane Π: 2x+y-2z=6.
• L2 meets Π at Q.
• R: foot of perpendicular from P to Π.
• ℓ = length PQ.
• A = area of triangle PQR.
• Find 4ℓ² A².

We need to compute ℓ² and A², then compute 4ℓ² A².

Approach:

1. Find direction vector of L1 by cross product of normals of given planes.
2. Parametric equation of L2: P + t * d.
3. Find intersection Q of L2 with Π by solving for t.
4. Compute ℓ = distance between P and Q = |t| * |d|.
5. Find foot R of perpendicular from P to Π.
6. Compute area A of triangle PQR using cross product: A = 0.5 * |(Q-P) × (R-P)|.
7. Compute 4ℓ² A².

Let's implement in Python using sympy for exact rational arithmetic.

Steps in code:

• Define variables x, y, z.
• Normals n1 = (2,3,1), n2 = (1,2,1).
• Direction d = cross(n1, n2).
• L2: (x,y,z) = (2, -1, 3) + t*d.
• Substitute into Π: 2x + y - 2z = 6, solve for t.
• Compute Q.
• ℓ = distance between P and Q = |t| * |d|.
• For foot R: formula for projection of point P onto plane Π. Plane: 2x + y - 2z = 6, normal n = (2,1,-2). R = P - ((2P[0] + P[1] - 2P[2] - 6) / (2^2+1^2+(-2)^2)) * n.
• Compute vectors Q-P and R-P.
• Area A = 0.5 * |cross(Q-P, R-P)|.
• Compute 4 * ℓ2 * A2.

Let's do it.

[/PYTHON] [PYTHON] import sympy as sp

Define variables

x, y, z = sp.symbols('x y z')

Normals of the two planes for L1

n1 = sp.Matrix([2, 3, 1]) n2 = sp.Matrix([1, 2, 1])

Direction vector of L1

d = n1.cross(n2) print("Direction vector d of L1:", d)

Point P

P = sp.Matrix([2, -1, 3])

Parametric equation for L2: P + t*d

t = sp.symbols('t') L2 = P + t * d print("Parametric L2:", L2)

Plane Π: 2x + y - 2z = 6

plane_eq = 2x + y - 2z - 6

Substitute L2 into plane equation

eq_sub = plane_eq.subs({x: L2[0], y: L2[1], z: L2[2]}) print("Equation after substitution:", eq_sub)

Solve for t

t_val = sp.solve(eq_sub, t) print("t value:", t_val)

Intersection point Q

t_val = t_val[0] Q = L2.subs(t, t_val) print("Point Q:", Q)

Compute distance ℓ = |t| * |d|

d_norm = sp.sqrt(d.dot(d)) ell = sp.Abs(t_val) * d_norm print("ℓ (distance PQ):", ell)

Foot R of perpendicular from P to plane Π

n = sp.Matrix([2, 1, -2]) plane_const = 6

Distance factor

d_factor = (P[0]*n[0] + P[1]*n[1] + P[2]*n[2] - plane_const) / (n.dot(n)) R = P - d_factor * n print("Foot R:", R)

Vectors Q-P and R-P

v1 = Q - P v2 = R - P

Area A = 0.5 * |cross(v1, v2)|

cross_prod = v1.cross(v2) area = sp.Rational(1, 2) * sp.sqrt(cross_prod.dot(cross_prod)) print("Area A:", area)

Compute 4 * ℓ² * A²

ell_sq = ell2 A_sq = area2 result = 4 * ell_sq * A_sq print("4 * ℓ² * A²:", result) print("Simplified result:", sp.simplify(result)) [/PYTHON]

[THOUGHT] The Python output gives the simplified result 511758. That is the value of $4\ell^2 A^2$. I will provide the final answer.

$\boxed{511758}$