
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 the binary operations $\clubsuit$ and $\spadesuit$ defined as $\clubsuit$ : $a^{\log_5(b)}$ and $\spadesuit$ : $a^{\frac{1}{\log_5(b)}}$ for real numbers $a$ and $b$ where these expressions are defined, a sequence $(b_n)$ is defined recursively as $b_4 = 4 \spadesuit 2$ and $b_n = (n \spadesuit (n-2)) \clubsuit b_{n-1}$ for all integers $n \geq 5$. Find $\log_5(b_{2023})$ to the nearest integer.

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