Five coins are placed in a line on a table. At the start, the coins in the 1st and 2nd positions show heads, and the coins in the remaining 3 positions show tails. Using these 5 coins and one die, the following trial is performed. Roll the die once. If the result is $k$, if $k \leq 5$, flip the coin in the $k$-th position once and place it back, if $k = 6$, flip all coins once and place them back. After repeating this trial 3 times, what is the probability that all 5 coins show heads? Express the answer as $\frac{q}{p}$. What is the value of $p + q$? (Given: $p$ and $q$ are coprime natural numbers.) [4 points]
Five coins are placed in a line on a table. At the start, the coins in the 1st and 2nd positions show heads, and the coins in the remaining 3 positions show tails.\\
Using these 5 coins and one die, the following trial is performed.\\
Roll the die once. If the result is $k$,\\
if $k \leq 5$, flip the coin in the $k$-th position once and place it back,\\
if $k = 6$, flip all coins once and place them back.\\
After repeating this trial 3 times, what is the probability that all 5 coins show heads? Express the answer as $\frac{q}{p}$. What is the value of $p + q$? (Given: $p$ and $q$ are coprime natural numbers.) [4 points]