\( \vec{x} \) and \( \vec{y} \) are two non zero, non-collinear vectors
\( \mathrm{P} \) satisfying \( \left[(a-3) \alpha^{2}+(b-4) \alpha+(c-1)\right] \overrightarrow{\mathrm{x}}+ \)
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\[
\begin{array}{l}
{\left[(\mathrm{a}-3) \beta^{2}+(\mathrm{b}-4) \beta+(\mathrm{c}-1)\right] \overrightarrow{\mathrm{y}}+} \\
{\left[(\mathrm{a}-3) \gamma^{2}+(\mathrm{b}-4) \gamma+(\mathrm{c}-1)\right](\overrightarrow{\mathrm{x}} \times \overrightarrow{\mathrm{y}})=0 \text { (where }}
\end{array}
\]
\( \alpha, \beta, \gamma \) are three distinct numbers), then the value of \( \frac{\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}}{4} \) is equal to
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