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⚡ AZUCATION SHORTCUT:
Total coins remains constant. Ratio 1: $17+7=24$. Ratio 2: $37+20=57$.
LCM(24, 57) = 456. Adjust ratios to sum to 456:
Initial: $323:133$ | Final: $296:160$.
Change in A: $323 - 296 = 27$ units. Given 27 units = 108 $\implies$ 1 unit = 4.
Total = $456 \times 4 = 1824$. Equal distribution = 912 each.

Step 1: Set up the initial variables
Let the number of coins in box A and box B be $17x$ and $7x$ respectively.

Step 2: Apply the first shift
108 coins are moved from A to B:
$\frac{17x - 108}{7x + 108} = \frac{37}{20}$
$20(17x - 108) = 37(7x + 108)$
$340x - 2160 = 259x + 3996$
$81x = 6156 \implies \mathbf{x = 76}$.

Step 3: Find current counts
After the first shift:
Box A = $17(76) - 108 = 1292 - 108 = \mathbf{1184}$ coins.
Box B = $7(76) + 108 = 532 + 108 = \mathbf{640}$ coins.
Total coins = $1184 + 640 = 1824$.

Step 4: Calculate further shift for 1:1 ratio
For a $1:1$ ratio, both boxes must have $\frac{1824}{2} = 912$ coins.
Coins to be shifted from A to B = Current A - Target A
$= 1184 - 912 = \mathbf{272}$.

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