Look and Say, Look and Say, Look and Say

$1,\ 11,\ 21,\ 1211,\ 111221,\ 312211,\ 13112221,\ \ldots$

In the above sequence with initial term 1, each subsequent term is generated by reading off the digits of the previous term, i.e. by counting from left to right the number of repeated digits. For example,

• 1 is read off as "one 1" or 11.
• 11 is read off as "two 1s" or 21.
• 21 is read off as "one 2, then one 1" or 1211.
• 1211 is read off as "one 1, one 2, then two 1s" or 111221.

How many times does the digit 3 appear in the $32^\text{nd}$ term of the sequence?