[THOUGHT] We need to compute S and then K. S involves:

• I: Region for z: disk with radius R and center c.
• II(2): I₁ = ∮_C f(z) dz, C: |z+1|=2. Also R_in = sum of residues of f(z) over poles inside that circle.
• III(2): I₂ = ∫_0^∞ sin x / x dx.

We need to compute:

1. From I: Find R and c from inequality: z\bar{z} + √2(z + \bar{z}) + 3i(z - \bar{z}) + 2 ≤ 0. Write z = x + iy, then \bar{z} = x - iy. Compute: z\bar{z} = x^2 + y^2. z + \bar{z} = 2x. z - \bar{z} = 2iy. So inequality: (x^2 + y^2) + √2 * 2x + 3i * (2iy) + 2 ≤ 0. Wait careful: 3i(z - \bar{z}) = 3i * (2iy) = 3i * 2iy = 6 i^2 y = -6y. So expression: x^2 + y^2 + 2√2 x - 6y + 2 ≤ 0. Complete squares: (x^2 + 2√2 x) + (y^2 - 6y) + 2 ≤ 0. For x: x^2 + 2√2 x = (x + √2)^2 - 2. For y: y^2 - 6y = (y - 3)^2 - 9. Then total: (x + √2)^2 - 2 + (y - 3)^2 - 9 + 2 ≤ 0 → (x + √2)^2 + (y - 3)^2 - 9 ≤ 0. Wait compute: -2 -9 +2 = -9. Yes. So (x + √2)^2 + (y - 3)^2 ≤ 9. Thus center c = -√2 + 3i? Actually x-coordinate: -√2, y-coordinate: 3. So c = -√2 + 3i. Radius R = sqrt(9) = 3.

2. From II: f(z) = (z^2 - 2) / ((z^2 + 2i) z^2). Poles: denominator zeros: (z^2 + 2i) z^2 = 0. So z^2 = 0 → z=0 (double pole? Actually z^2 factor gives pole order 2 at z=0). And z^2 + 2i = 0 → z^2 = -2i = 2 e^{-iπ/2}? Actually -2i = 2 * (-i) = 2 e^{-iπ/2}. So z = ±√(2) e^{-iπ/4} = ±√2 (cos π/4 - i sin π/4) = ±√2 (1/√2 - i/√2) = ±(1 - i). Wait compute: √2 e^{-iπ/4} = √2 (cos π/4 - i sin π/4) = √2 (√2/2 - i√2/2) = 1 - i. So poles at z = 1 - i and z = -1 + i? Actually ±(1 - i): second root is -(1 - i) = -1 + i. So poles: z=0 (order 2), z=1-i (order 1), z=-1+i (order 1). Need residues at each pole.

   Compute residues: For simple poles: residue = lim_{z→z0} (z - z0) f(z). For double pole at 0: residue = coefficient of 1/z in Laurent expansion.

   Let's compute using sympy.

   Also need I₁ = ∮_C f(z) dz = 2πi * sum of residues inside C. C: |z+1|=2, center at -1, radius 2. Determine which poles lie inside: compute distances from -1:

   • z=0: distance |0+1|=1 <2 → inside.
   • z=1-i: distance |1-i +1| = |2 - i| = sqrt(4+1)=√5 ≈2.236 >2? Actually √5 ≈2.236 >2 → outside.
   • z=-1+i: distance |-1+i+1| = |i| = 1 <2 → inside. So poles inside: z=0 and z=-1+i. Sum of residues inside = Res at 0 + Res at -1+i = R_in.

   Compute I₁ = 2πi * R_in.

3. From III(2): I₂ = ∫_0^∞ sin x / x dx = π/2 (known result). But we can compute via contour integration if needed, but we can trust known value π/2.

   However, we need exact numeric value for S and K. S involves (|I₁|/I₂)^3. I₁ is complex? Actually I₁ is integral of f(z) dz, which is complex number. |I₁| is its magnitude.

   So compute numeric values.

   Steps:

   • Compute residues using sympy.
   • Compute R_in = Res(0) + Res(-1+i).
   • Compute I₁ = 2πi * R_in.
   • Compute |I₁|.
   • Compute I₂ = π/2.
   • Compute R = 3, c = -√2 + 3i, so |c| = sqrt( (√2)^2 + 3^2 ) = sqrt(2+9)=√11.
   • Then R^2 + |c|^2 = 9 + 11 = 20.
   • So (R^2 + |c|^2)^2 = 400.
   • Compute |I₁|/I₂ = |I₁| / (π/2) = 2|I₁|/π.
   • Cube: (2|I₁|/π)^3 = 8 |I₁|^3 / π^3.
   • Multiply by 400: S = 400 * 8 |I₁|^3 / π^3 * 1/|R_in| = 3200 |I₁|^3 / (π^3 |R_in|).
   • Then K = 999 S^2 + 7.

   Need numeric values. Let's compute with Python. [/THOUGHT] [PYTHON] import sympy as sp import cmath import math

Define symbols

z = sp.symbols('z')

Define f(z)

f = (z2 - 2) / ((z2 + 2*sp.I) * z**2) print("f(z) =", f)

Find poles

denom = sp.denom(f) print("Denominator:", denom)

Solve denominator = 0

poles = sp.solve(denom, z) print("Poles:", poles)

Also get multiplicities? Let's compute residues directly.

Compute residues at each pole

Pole at z=0 (order 2)

res0 = sp.residue(f, z, 0) print("Residue at z=0:", res0, "=", complex(res0.evalf()))

Pole at z = 1 - i

z1 = 1 - 1j # Python uses j for imaginary res1 = sp.residue(f, z, 1 - sp.I) print("Residue at z=1-i:", res1, "=", complex(res1.evalf()))

Pole at z = -1 + i

z2 = -1 + 1j res2 = sp.residue(f, z, -1 + sp.I) print("Residue at z=-1+i:", res2, "=", complex(res2.evalf()))

Compute R_in = sum of residues inside C: z=0 and z=-1+i

R_in = res0 + res2 print("R_in = Res0 + Res(-1+i) =", R_in, "=", complex(R_in.evalf()))

Compute I1 = 2πi * R_in

I1 = 2 * sp.pi * sp.I * R_in print("I1 = 2πi * R_in =", I1, "=", complex(I1.evalf()))

Compute |I1|

I1_complex = complex(I1.evalf()) abs_I1 = abs(I1_complex) print("|I1| =", abs_I1)

I2 = π/2

I2 = sp.pi / 2 print("I2 = π/2 =", float(I2.evalf()))

Compute R and c from part I

R = 3 c = -math.sqrt(2) + 3j abs_c = abs(c) print("R =", R) print("c =", c) print("|c| =", abs_c)

Compute (R^2 + |c|^2)^2

term1 = R2 + abs_c2 print("R^2 + |c|^2 =", term1) term1_sq = term1**2 print("(R^2 + |c|^2)^2 =", term1_sq)

Compute |I1|/I2

ratio = abs_I1 / float(I2.evalf()) print("|I1|/I2 =", ratio) ratio_cubed = ratio**3 print("(|I1|/I2)^3 =", ratio_cubed)

Compute |R_in|

abs_R_in = abs(complex(R_in.evalf())) print("|R_in| =", abs_R_in)

Compute S

S = term1_sq * ratio_cubed / abs_R_in print("S =", S)

Compute K = 999 S^2 + 7

K = 999 * S**2 + 7 print("K = 999 S^2 + 7 =", K) print("K as integer:", int(K)) [/PYTHON]

[THOUGHT] The Python computation gives S = 6400.0 and K = 40919040007.0. The integer value is 40919040007. [/THOUGHT] \boxed{40919040007}