Find the sum of $\unicode{x1F34E} + \unicode{x1F34C} + \unicode{x1F34D}$ for all triples of positive integers $\unicode{x1F34E}, \unicode{x1F34C}, \unicode{x1F34D}$ which satisfy

$$\frac {\unicode{x1F34E}} {\unicode{x1F34C} + \unicode{x1F34D}} + \frac {\unicode{x1F34C}} {\unicode{x1F34D} + \unicode{x1F34E}} + \frac {\unicode{x1F34D}} {\unicode{x1F34E} + \unicode{x1F34D}} = 4$$

and $\unicode{x1F34E} + \unicode{x1F34C} + \unicode{x1F34D} \le 10^7$.

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