Find the least natural number $n$, which has at least 6 different divisors $1 = d_1 < d_2 < d_3 < d_4 < d_5 < d_6 < \ldots$, such that:
1. $d_3 + d_4 = d_5 + 6$
2. $d_4 + d_5 = d_6 + 7$

Let's think step by step and output the final answer within \boxed{}.
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