
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
Let $a\neq 0$, $x$, and $y$ be positive integers. A new operation is defined as $a\oplus x=a^{x}$ (if there are parentheses, the calculation inside the parentheses is done first). For example, $2\oplus 2=2^{2}=4$, $4\oplus \left(m+1\right)=4^{m+1}$.

$(1)$ If $2\oplus n=64$, then $n=$______; if $n\oplus 2=64$, then $n=$______.

$(2)$ If $\left(2\oplus x\right)\left(2\oplus 2y\right)=8$ and $\left(3\oplus x\right)\left(3\oplus y\right)=9$, find the values of $x$ and $y$.

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