
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 \(\{a_{n}\}\) and \(\{b_{n}\}\) be sequences defined recursively by \(a_{0}=2\), \(b_{0}=2\), and
\[a_{n+1}=a_{n} \sqrt{1+a_{n}^{2}+b_{n}^{2}} - b_{n},\]
\[b_{n+1}=b_{n} \sqrt{1+a_{n}^{2}+b_{n}^{2}} + a_{n}.\]
Find the ternary (base 3) representation of \(a_{4}\) and \(b_{4}\).

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