
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 $\diamondsuit$ and $\heartsuit$ by \[a \, \diamondsuit \, b = a^{\log_{10}(b)} \qquad \text{and} \qquad a  \, \heartsuit \, b = a^{\frac{1}{\log_{10}(b)}}\]for all real numbers $a$ and $b$ where these expressions are defined. The sequence $(b_n)$ is defined recursively by $b_4 = 4\, \heartsuit\, 3$ and \[b_n = (n\, \heartsuit\, (n-1)) \,\diamondsuit\, b_{n-1}\]for all integers $n \geq 5$. To the nearest integer, what is $\log_{10}(b_{2021})$?
A) 3
B) 4
C) 5
D) 6
E) 7

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