阅读材料:求$1+2+2^{2}+2^{3}+\ldots +2^{2023}+2^{2024}$的值.
解:设$S=1+2+2^{2}+2^{3}+\ldots +2^{2023}+2^{2024}$①,将等式①的两边同乘以$2$,
得$2S=2+2^{2}+2^{3}+2^{4}+\ldots +2^{2024}+2^{2025}$②,
用②$-$①得,$2S-S=2^{2025}-1$,
即$S=2^{2025}-1$.
所以,$1+2+2^{2}+2^{3}+\ldots +2^{2023}+2^{2024}=2^{2025}-1$.
请仿照此法计算:
$(1)$填空:$1+3+3^{2}+3^{3}+3^{4}+3^{5}=$______；
$(2)$求$1+7+7^{2}+7^{3}+\ldots +7^{2023}+7^{2024}$的值.