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In part 4.2 of this paper, the writers introduced a technique called 'Multi-step Modification', here I got some questions. How does the Multi-step Modification ensure that we will not change any condition in the first round (i.e. what we have changed in so called 'single step modification'.)? The writers also claimed that the order of modification matters (i.e. we must firstly correct $a_{5, \space 19}$, then $a_{5, \space 26}$ and so on), but I don't understand this.

Here is my own understanding on the first question and I am not sure if I got the right idea. We changed $m_0$ so that $a_1, \space d_1, \space c_1, \space b_1$ will be changed. For correction, we first calculate the new value of $a_1$, and use the new value to recompute $m_1, \space m_2, \space m_3, \space m_4$ so that the value of $d_1, \space c_1, \space b_1$ does not change. As showed in the C++ code segment.

Thanks for any help.

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