# Bringing A whole new level

For every positive integer $$g$$ there is a unique factorial base expansion $$(g_1,g_2,g_3,\ldots,g_m)$$, meaning that

$$g=1!\cdot g_1+2!\cdot g_2+3!\cdot g_3+\cdots+m!\cdot g_m$$, $$\text{where each}$$ $$g_i$$ is an integer, $$0\le g_i\le i$$ , and $$0<g_m$$.

Given that $$(g_1,g_2,g_3,\ldots,g_j)$$ is the factorial base expansion of $$16!-32!+48!-64!+\cdots+1968!-1984!+2000!$$, find the value of $$g_1-g_2+g_3-g_4+\cdots+(-1)^{j+1}g_j$$.

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