Consider the equation $3si{n}^{\text{'}}(x+y)+3sin(x+z)+4sin(y+z)=0.$ Find the values of $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{x}}$ and $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{y}}$ at the point $(-3,-3,-4).$

$\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{x}}{|}_{(-3,-3,-4)}||=$

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$($Simplify your answer. Type an exact answer, using radicals as needed.$)$