Determine all positive, composite integers $n$ that satisfy the following property: if the positive divisors of $n$ are $1=d_{1}<d_{2}<\cdots<d_{k}=n$, then $d_{i}$ divides $d_{i+1}+d_{i+2}$ for every $1 \leqslant i \leqslant k-2$. Please provide the minimum value of $n $that meets the criteria.

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