
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 arithmetic sequences $\left(a_n\right)$ and $\left(b_n\right)$ with integer terms where $a_1=2, b_1=3$ and differences $x < y$, calculate the largest value of $n$ for which the product $a_n b_n = 1638$.

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