
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Six distinct prime numbers are selected: 2, 3, 5, 7, 17, 19. These numbers are arranged to fill a 3x3 multiplication table, with three different primes across the top and the remaining three down the left side. Each cell of the table is the product of the corresponding row and column prime numbers. What is the largest possible sum of the nine entries in the 3x3 multiplication table?

\[
\begin{array}{c||c|c|c|}
\times & a & b & c \\ \hline \hline
d & & & \\ \hline
e & & & \\ \hline
f & & & \\ \hline
\end{array}
\]

Present the answer in LaTex format: \boxed{Your answer}
user