
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
Find the largest possible value of \( n \) in arithmetic sequences \( (a_n) \) with \( a_1 = 1 \) and \( (b_n) \) with \( b_1 = 3 \), and both having integer terms, such that the product \( a_n b_n = 2304 \) for some \( n \). Assume \( a_n = 1 + (n-1)d_a \) and \( b_n = 3 + (n-1)d_b \) where \( d_a \) and \( d_b \) are integers.
A) 12
B) 17
C) 19
D) 20
E) 23

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