
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
A line passes through the point $P(\frac{4}{3}, 2)$ and intersects the positive semi-axes of the $x$-axis and $y$-axis at points $A$ and $B$, respectively. Point $O$ is the origin. Determine whether there exists a line that simultaneously satisfies the following conditions:

(i) The perimeter of $\triangle AOB$ is $12$;

(ii) The area of $\triangle AOB$ is $6$. If such a line exists, find its equation; otherwise, explain the reason.

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