
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 binary operations $\clubsuit$ and $\spadesuit$ by
\[ a \clubsuit b = a^{\log_{5}(b)} \qquad \text{and} \qquad a \spadesuit b = a^{\frac{1}{\log_{5}(b)}} \]
for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(b_n)$ is defined recursively by $b_4 = 4 \spadesuit 3$ and
\[ b_n = (n \spadesuit (n-1)) \clubsuit b_{n-1} \]
for all integers $n \geq 5$. To the nearest integer, what is $\log_{5}(b_{2023})$?
A) 3
B) 4
C) 5
D) 6

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