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$\begin{cases} \begin{aligned}a_1&=1\\ a_{n+1}&=2a_{n}+n2^n, \quad n\ge 1 \end{aligned}\end{cases}$

Let $\left \{a_k\right \}$ be a sequence satisfying the above condition. Find the minimum $n$ such that $a_n > 2^{100}$.

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