The answer is 4031.0.

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$\cot C=2015 (\cot A+\cot B) \\ \Rightarrow \frac {\cot C} {\cot A+\cot B} =2015 \\ \Rightarrow \frac{\left ( \frac{\cos C}{\sin C} \right )}{\left ( \frac{\cos A}{\sin A}+\frac{\cos B}{\sin B} \right )}=2015$

$\Rightarrow \frac{\frac{\cos C}{\sin C}}{\frac{\cos A}{\sin A}+\frac{\cos B}{\sin B}}=2015$

$\Rightarrow \frac{\cos C}{\sin C}\times \frac{\sin A\cdot \sin B}{\sin (A+B)}=2015$

$\Rightarrow \frac{\sin A\cdot \sin B\cdot \cos C}{\sin^{2}C}=2015$

$\Rightarrow \frac{ab}{c^{2}}\cdot \frac{a^{2}+b^{2}-c^{2}}{2ab}=2015$

(note that the above step is due to $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ )

$\Rightarrow \frac{a^{2}+b^{2}-c^{2}}{2c^{2}}=2015$

$\Rightarrow \frac{t-1}{2}=2015$

$\Rightarrow \boxed{t=4031}.$