
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
Given $a_{1}, a_{2}, \cdots, a_{n}, b_{1}, b_{2}, \cdots, b_{n}$ be $2n$ distinct real numbers, and let $M$ be an $n$-order extended matrix where the element in the $i$-th row and $j$-th column is $a_{i}+b_{j}$. If the product of the elements in each column is 1, calculate the product of the elements in each row.

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