Let $k$ be a positive integer. A sequence $a_0, a_1, \ldots, a_n$, where $n > 0$, of positive integers satisfies the following conditions:

1. $a_0 = a_n = 1$;
2. $2 \leq a_i \leq k$ for each $i = 1, 2, \ldots, n-1$;
3. For each $j = 2, 3, \ldots, k$, the number $j$ appears $\phi(j)$ times in the sequence $a_0, a_1, \ldots, a_n$, where $\phi(j)$ is the number of positive integers that do not exceed $j$ and are coprime to $j$;
4. For any $i = 1, 2, \ldots, n-1$, $\gcd(a_i, a_{i-1}) = 1 = \gcd(a_i, a_{i+1})$, and $a_i$ divides $a_{i-1} + a_{i+1}$.

Suppose there is another sequence $b_0, b_1, \ldots, b_n$ of integers such that $\frac{b_{i+1}}{a_{i+1}} > \frac{b_i}{a_i}$ for all $i = 0, 1, \ldots, n-1$. Find the minimum value of $b_n - b_0$.

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