For a natural number $m$, blocks in the shape of identical cubes are stacked with 1 block in column 1, 2 blocks in column 2, 3 blocks in column 3, $\cdots$, and $m$ blocks in column $m$. The following trial is repeated until there are no columns with an even number of blocks remaining. For each column with an even number of blocks, remove from that column a number of blocks equal to $\frac { 1 } { 2 }$ of the number of blocks in that column. Let $f ( m )$ be the sum of the number of blocks remaining in columns 1 through $m$ after all block removal trials are completed. For example, $f ( 2 ) = 2 , f ( 3 ) = 5 , f ( 4 ) = 6$. $$\lim _ { n \rightarrow \infty } \frac { f \left( 2 ^ { n + 1 } \right) - f \left( 2 ^ { n } \right) } { f \left( 2 ^ { n + 2 } \right) } = \frac { q } { p }$$ Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]
For a natural number $m$, blocks in the shape of identical cubes are stacked with 1 block in column 1, 2 blocks in column 2, 3 blocks in column 3, $\cdots$, and $m$ blocks in column $m$. The following trial is repeated until there are no columns with an even number of blocks remaining.
For each column with an even number of blocks, remove from that column a number of blocks equal to $\frac { 1 } { 2 }$ of the number of blocks in that column.
Let $f ( m )$ be the sum of the number of blocks remaining in columns 1 through $m$ after all block removal trials are completed. For example, $f ( 2 ) = 2 , f ( 3 ) = 5 , f ( 4 ) = 6$.
$$\lim _ { n \rightarrow \infty } \frac { f \left( 2 ^ { n + 1 } \right) - f \left( 2 ^ { n } \right) } { f \left( 2 ^ { n + 2 } \right) } = \frac { q } { p }$$
Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]