
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 
\[a \, \diamondsuit \, b = a^{\log_{10}(b)} \quad \text{and} \quad a \, \heartsuit \, b = a^{\frac{1}{\log_{10}(b)}}\]
for all real numbers $a$ and $b$ where these expressions are defined. The sequence $(a_n)$ is defined recursively by $a_3 = 5 \, \heartsuit \, 3$ and 
\[a_n = (n \, \heartsuit \, (n-1)) \, \diamondsuit \, (a_{n-1}+1)\]
for all integers $n \geq 4$. To the nearest integer, what is $\log_{10}(a_{2020})$?
A) 2
B) 3
C) 4
D) 5

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