You are a helpful assistant. When the user asks a question, your response must include two parts: first, the reasoning process enclosed in <think>...</think> tags, then the final answer enclosed in <answer>...</answer> tags.Please provide a clear, concise response within <answer> </answer> tags that directly addresses the question.
system
<image>The $27$ cells of a $3 \times 9$ grid are filled in using the numbers $1$ through $9$ so that each row contains $9$ different numbers, and each of the three $3 \times 3$ blocks heavily outlined in the example below contains $9$ different numbers, as in the first three rows of a Sudoku puzzle.

\[
\begin{array}{|c|c|c||c|c|c||c|c|c|}
\hline
4 & 2 & 8 & 9 & 6 & 3 & 1 & 7 & 5 \\
\hline
3 & 7 & 9 & 5 & 2 & 1 & 6 & 8 & 4 \\
\hline
5 & 6 & 1 & 8 & 4 & 7 & 9 & 2 & 3 \\
\hline
\end{array}
\]
The number of different ways to fill such a grid can be written as $p^a \cdot q^b \cdot r^c \cdot s^d$ where $p$, $q$, $r$, and $s$ are distinct prime numbers and $a$, $b$, $c$, $d$ are positive integers. Find $p \cdot a + q \cdot b + r \cdot c + s \cdot d$. You FIRST think about the reasoning process as an internal monologue and then provide the final answer. The reasoning process MUST BE enclosed within <think> </think> tags. Please answer the final answer within the <answer> </answer> tags.
user
81
The $27$ cells of a $3 \times 9$ grid are filled in using the numbers $1$ through $9$ so that each row contains $9$ different numbers, and each of the three $3 \times 3$ blocks heavily outlined in the example below contains $9$ different numbers, as in the first three rows of a Sudoku puzzle.

\[
\begin{array}{|c|c|c||c|c|c||c|c|c|}
\hline
4 & 2 & 8 & 9 & 6 & 3 & 1 & 7 & 5 \\
\hline
3 & 7 & 9 & 5 & 2 & 1 & 6 & 8 & 4 \\
\hline
5 & 6 & 1 & 8 & 4 & 7 & 9 & 2 & 3 \\
\hline
\end{array}
\]
The number of different ways to fill such a grid can be written as $p^a \cdot q^b \cdot r^c \cdot s^d$ where $p$, $q$, $r$, and $s$ are distinct prime numbers and $a$, $b$, $c$, $d$ are positive integers. Find $p \cdot a + q \cdot b + r \cdot c + s \cdot d$. You FIRST think about the reasoning process as an internal monologue and then provide the final answer. The reasoning process MUST BE enclosed within <think> </think> tags. Please answer the final answer within the <answer> </answer> tags.
val