
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 that the binary operations $\diamondsuit$ and $\heartsuit$ are defined by $a \, \diamondsuit \, b = a^{\log_{5}(b)}$ and $a \, \heartsuit \, b = a^{\frac{1}{\log_{5}(b)}}$ respectively, and the sequence $(a_n)$ is defined recursively by $a_4 = 4\, \heartsuit\, 3$ and $a_n = (n\, \heartsuit\, (n-1)) \,\diamondsuit\, a_{n-1}$ for all integers $n \geq 5$, find $\log_{5}(a_{2024})$.

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