Not So Flat After All

Every positive integer $v$ can be written as $v = p_1^{a_1} \cdot p_2^{a_2} \cdots p_n^{a_n}$, where each $p_i$ is a prime and every $a_i \ge 0$. For example, $24 = 2^3 \cdot 3^1$.

Pick two distinct primes $p_1 \ne p_2$. Consider a two-dimensional plane on which the exponent of $p_1$ is the x-coordinate and the exponent of $p_2$ is the y-coordinate. Any number of the form $p_1^{a_1} \cdot p_2^{a_2}$ is then the point $(a_1, a_2)$.

This idea extends to any $N$-dimensional space in which each of the $N$ axes is assigned a distinct prime. Every such space has a unique set of primes, which we call the Space Identification Set $S$; its size $|S|$ equals $N$. Any number that is a product of primes taken only from $S$ (each raised to an exponent $\ge 0$) can be plotted in this $|S|$-dimensional space. Naturally, any number plottable in space $A$ is also plottable in space $B$ whenever $S_A \subseteq S_B$.

The distance between two points is the number of unit steps needed to travel from one to the other along the grid lines, where every move is parallel to a single axis. This equals the Manhattan (L1) distance between the two coordinate vectors. For instance, $168 = 2^3 \cdot 3 \cdot 7$ and $882 = 2 \cdot 3^2 \cdot 7^2$ are at distance $|3-1| + |1-2| + |1-2| = 4$.

Given two positive integers, determine the minimum size of a space in which both numbers can be plotted, and the distance between the two numbers in that space.

The input contains one or more test cases. Each test case is a single line with two positive integers $A$ and $B$ ($0 < A, B < 1{,}000{,}000$) satisfying $A \cdot B > 1$. The input ends with a line containing two zeros, which must not be processed.

For each test case, print one line in the form:

k. X:D

where $k$ is the test case number (starting from 1), $X$ is the minimum size of a space in which both $A$ and $B$ can be plotted, and $D$ is the distance between the two numbers in that space.