
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 three lines $l_{1}, l_{2}, l_{3}$ in a plane not passing through the origin, with equations:
$$
\begin{array}{l}
l_{1}: a_{1} x + b_{1} y + c_{1} = 0, \\
l_{2}: a_{2} x + b_{2} y + c_{2} = 0, \\
l_{3}: a_{3} x + b_{3} y + c_{3} = 0.
\end{array}
$$
A moving line $l$ that passes through the origin intersects these lines at points $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right), C\left(x_{3}, y_{3}\right)$. Let $\left|\overrightarrow{OA}_{i}\right| = \rho_{i} (i=1, 2, 3)$. Point $P$ lies on line $l$ with $\left|\overrightarrow{OP}\right| = \rho$, and it satisfies:
$$
\frac{1}{\rho_{1}} + \frac{1}{\rho_{2}} + \frac{1}{\rho_{3}} = \frac{1}{\rho}.
$$
Find the equation of the locus of the moving point $P(x, y)$.

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