
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]


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Given that $a > 1$, there exists exactly one constant $c$ such that for any $x \in [a, 3a]$, there exists a $y \in [a, a^3]$ satisfying the equation $\log_a x + \log_a y = c$. The set of values of $a$ that satisfy this condition is __________.

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