$X$、$Y$ 和 $Z$ 是两位数。$X$、$Y$ 和 $Z$ 的平均数是 56。$X$ 是 $Y$ 的 $\frac{2}{3}$。求 $X$ 的最小可能值。 答：___

1. Find the total sum of $X$, $Y$, and $Z$.

$\text{Total sum} = 56 \times 3 = 168$
$X + Y + Z = 168$

2. Express the sum in terms of $X$ and $Z$.

Since $X = \frac{2}{3}Y$, we have $Y = \frac{3}{2}X$.
Substitute $Y$ into the sum equation:
$X + \frac{3}{2}X + Z = 168$
$\frac{5}{2}X + Z = 168$

3. Use the constraint on $Z$ to find the minimum value of $X$.

To find the smallest possible value of $X$, we must make $Z$ as large as possible.
Since $Z$ is a 2-digit number, its maximum value is $99$.
$\frac{5}{2}X + 99 \ge 168$
$\frac{5}{2}X \ge 69$
$X \ge 69 \times \frac{2}{5} = 27.6$

4. Determine the smallest valid integer for $X$.

Since $Y = \frac{3}{2}X$ must be an integer, $X$ must be an even number.
The smallest even integer greater than $27.6$ is $28$.
Let's check if $X = 28$ works:
$Y = \frac{3}{2}(28) = 42$ (a 2-digit number)
$Z = 168 - \frac{5}{2}(28) = 168 - 70 = 98$ (a 2-digit number)
Both $Y$ and $Z$ are valid 2-digit numbers.