Let $T = \frac{1}{4}x^{2} - \frac{1}{5}y^{2} + \frac{1}{6}z^{2}$ where $x, y, z$ are real numbers such that $1 \leq x, y, z \leq 4$ and $x - y + z = 4$. Find the smallest value of $10 \times T$.

Let's think step by step and output the final answer within \boxed{}.
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