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The prime symbol (′) is a common mathematical notation used to denote a variable, such as `a′` or `v′`, that is derived from or related to another variable (`a` or `v`) without needing a new name. This notation, read as 'a-prime' or 'v-prime', can signify a value after a transformation, a derivative, or a subsequent state. It is frequently combined with subscripts, as in `v_i'`, which represents the prime version of the i-th element `v_i`.

Prime Notation in Mathematics

In mathematical and scientific contexts, the notation `x′` (read as 'x-prime') exclusively signifies the derivative of the variable `x`.

An algorithm tracks the coordinates of three points, represented as `c_1`, `c_2`, and `c_3`. After one iteration of the algorithm, all three points are updated to new coordinates. Following standard mathematical convention, how would you represent the new coordinate of the second point (`c_2`)?

Applying Mathematical Notation for Updated Variables

A programmer is documenting a process to update a value. They write the following steps:
1. Let `v` be the original value.
2. A transformation is applied to `v` to get a new value, `v′`.
3. A second transformation is applied to `v′` to get the final value, `v′′`.

In this context, what is the role of `v′`?

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