
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
Define a sequence of numbers by \( b_1 = 5 \), \( b_2 = \frac{5}{13} \), and for \( n \geq 3 \),
\[ b_n = \frac{b_{n-2} \cdot b_{n-1}}{3b_{n-2} - b_{n-1}}. \]
Find \( b_{2023} \) expressed as \( \frac{p}{q} \), where \( p \) and \( q \) are relatively prime integers, and calculate \( p+q \).
A) 10100
B) 10113
C) 10115
D) 10120
E) 10205

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