To find the maximum possible value of the product $ xy $, we need to solve for $ x $ and $ y $ based on the given equations and then determine the product $ xy $.

First, solve the equation $ | x + 3 | = 4 $:

1. $ x + 3 = 4 $
$$
x = 4 - 3 = 1
$$

2. $ x + 3 = -4 $
$$
x = -4 - 3 = -7
$$

So, the possible values for $ x $ are $ x = 1 $ and $ x = -7 $.

Next, find the corresponding values of $ y $ using the equation $ y = 7 - | x - 2 | $:

1. For $ x = 1 $:
$$
y = 7 - | 1 - 2 | = 7 - | -1 | = 7 - 1 = 6
$$

2. For $ x = -7 $:
$$
y = 7 - | -7 - 2 | = 7 - | -9 | = 7 - 9 = -2
$$

Now, calculate the product $ xy $ for each pair $(x, y)$:

1. For $ x = 1 $ and $ y = 6 $:
$$
xy = 1 \cdot 6 = 6
$$

2. For $ x = -7 $ and $ y = -2 $:
$$
xy = -7 \cdot -2 = 14
$$

Therefore, the maximum possible value of the product $ xy $ is $ 14 $.

Question

To find the maximum possible value of the product $ xy $, we need to solve for $ x $ and $ y $ based on the given equations and then determine the product $ xy $.

First, solve the equation $ | x + 3 | = 4 $:

1. $ x + 3 = 4 $
   $$
   x = 4 - 3 = 1
   $$

2. $ x + 3 = -4 $
   $$
   x = -4 - 3 = -7
   $$

So, the possible values for $ x $ are $ x = 1 $ and $ x = -7 $.

Next, find the corresponding values of $ y $ using the equation $ y = 7 - | x - 2 | $:

1. For $ x = 1 $:
   $$
   y = 7 - | 1 - 2 | = 7 - | -1 | = 7 - 1 = 6
   $$

2. For $ x = -7 $:
   $$
   y = 7 - | -7 - 2 | = 7 - | -9 | = 7 - 9 = -2
   $$

Now, calculate the product $ xy $ for each pair $(x, y)$:

1. For $ x = 1 $ and $ y = 6 $:
   $$
   xy = 1 \cdot 6 = 6
   $$

2. For $ x = -7 $ and $ y = -2 $:
   $$
   xy = -7 \cdot -2 = 14
   $$

Therefore, the maximum possible value of the product $ xy $ is $ 14 $.

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