
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 $\star$ and $\circ$ by 
\[
a \star b = a^{\log_{5}(b)} \quad \text{and} \quad a \circ b = a^{\log_{b}(5)}
\]
for real numbers $a$ and $b$ where these operations are defined. A sequence $(b_n)$ is defined recursively by $b_3 = 5 \circ 2$ and 
\[
b_n = (n \circ (n-1)) \star b_{n-1}
\]
for all integers $n \geq 4$. Find the integer closest to $\log_{5}(b_{2023})$.
A) 3
B) 4
C) 5
D) 6

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