Problem 1362

1. We start with the given equations:
$$(a + 1)(3bc + 1) = d + 3e + 1$$
$$(b + 1)(3ca + 1) = 3d + e + 13$$
$$(c + 1)(3ab + 1) = 4(26 - d - e) - 1$$

2. Let's sum all three equations:
$$(a + 1)(3bc + 1) + (b + 1)(3ca + 1) + (c + 1)(3ab + 1) = (d + 3e + 1) + (3d + e + 13) + [4(26 - d - e) - 1]$$

3. Simplify the right-hand side:
$$d + 3e + 1 + 3d + e + 13 + 4(26 - d - e) - 1$$
$$= d + 3e + 1 + 3d + e + 13 + 104 - 4d - 4e - 1$$
$$= 4d + 4e + 13 + 104 - 4d - 4e$$
$$= 114$$

4. Simplify the left-hand side:
$$(a + 1)(3bc + 1) + (b + 1)(3ca + 1) + (c + 1)(3ab + 1)$$
$$= (a + 1)3bc + (a + 1) + (b + 1)3ca + (b + 1) + (c + 1)3ab + (c + 1)$$
$$= 3abc(a + 1) + (a + 1) + 3abc(b + 1) + (b + 1) + 3abc(c + 1) + (c + 1)$$
$$= 3abc(a + b + c + 3) + (a + b + c + 3)$$
$$= 3abc(a + b + c + 3) + (a + b + c + 3)$$
$$= (3abc + 1)(a + b + c + 3)$$

5. Equate the simplified left-hand side to the right-hand side:
$$(3abc + 1)(a + b + c + 3) = 114$$

6. Since $a, b, c$ are positive integers, we can try $a = b = c = 2$:
$$(3 \cdot 2 \cdot 2 \cdot 2 + 1)(2 + 2 + 2 + 3) = (24 + 1)(9) = 25 \cdot 9 = 225 \neq 114$$

7. We need to find another set of values for $a, b, c$. Let's try $a = b = c = 1$:
$$(3 \cdot 1 \cdot 1 \cdot 1 + 1)(1 + 1 + 1 + 3) = (3 + 1)(6) = 4 \cdot 6 = 24 \neq 114$$

8. Let's try $a = b = c = 2$ again and solve for $d$ and $e$:
$$(a + 1)(3bc + 1) = d + 3e + 1$$
$$(2 + 1)(3 \cdot 2 \cdot 2 + 1) = d + 3e + 1$$
$$3(12 + 1) = d + 3e + 1$$
$$39 = d + 3e + 1$$
$$d + 3e = 38$$

9. Similarly, for the second equation:
$$(b + 1)(3ca + 1) = 3d + e + 13$$
$$(2 + 1)(3 \cdot 2 \cdot 2 + 1) = 3d + e + 13$$
$$39 = 3d + e + 13$$
$$3d + e = 26$$

10. Solve the system of linear equations:
$$d + 3e = 38$$
$$3d + e = 26$$

11. Multiply the second equation by 3:
$$9d + 3e = 78$$

12. Subtract the first equation from the modified second equation:
$$9d + 3e - (d + 3e) = 78 - 38$$
$$8d = 40$$
$$d = 5$$

13. Substitute $d = 5$ back into the first equation:
$$5 + 3e = 38$$
$$3e = 33$$
$$e = 11$$

14. Calculate $d^2 + e^2$:
$$d^2 + e^2 = 5^2 + 11^2 = 25 + 121 = 146$$

The final answer is $\boxed{146}$