[PYTHON] import math

Equate coefficients to parse the derived circle equation

h_center = -1/3 k_center = 10/3 radius_squared = 8 / 9

val = 9 * (h_center + k_center) * radius_squared print(round(val)) [/PYTHON]

\boxed{24}[PYTHON] import math

Equate coefficients to parse the derived circle equation

h_center = -1/3 k_center = 10/3 radius_squared = 8 / 9

val = 9 * (h_center + k_center) * radius_squared print(round(val)) [/PYTHON]

\boxed{24}

[THOUGHT] Using the derived circle equation, center is (−1/3,10/3) and radius squared is 8/9. Compute 9*(sum of center coordinates) * radius squared: (9 \cdot ( (-1/3)+(10/3) ) \cdot (8/9)). Simplifying yields 24.

Final Answer: 24

[THOUGHT] Final calculation comes down to 3*8 =24 [/THOUGHT]

\boxed{24}

# Validation via computational check to ensure correctness. From previous analysis:

h_center = -1/3
k_center = 10/3
radius_squared = 8/9

result = 9 * (h_center + k_center) * radius_squared
print(int(result))

\boxed{24}[THOUGHT] Using the derived circle equation, center is (−1/3,10/3) and radius squared is 8/9. Compute 9*(sum of center coordinates) * radius squared: (9 \cdot ( (-1/3)+(10/3) ) \cdot (8/9)). Simplifying yields 24.

Final Answer: 24

[THOUGHT] Final calculation comes down to 3*8 =24 [/THOUGHT]

\boxed{24}

# Validation via computational check to ensure correctness. From previous analysis:

h_center = -1/3
k_center = 10/3
radius_squared = 8/9

result = 9 * (h_center + k_center) * radius_squared
print(int(result))

\boxed{24}

[THOUGHT] Python confirmed that the calculated value for 9(x₀ + y₀)r² is 24 based on the derived circle parameters. This result is accurate per the computations done earlier. [/THOUGHT]

\boxed{24}[THOUGHT] Python confirmed that the calculated value for 9(x₀ + y₀)r² is 24 based on the derived circle parameters. This result is accurate per the computations done earlier. [/THOUGHT]

\boxed{24}