
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 $P$ be any point in the first quadrant on the ellipse $C: \frac{x^{2}}{5} + y^{2} = 1$. $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse, respectively. The lines $P F_{1}$ and $P F_{2}$ intersect the ellipse $C$ at points $M$ and $N$, respectively. Given that $\overrightarrow{P F_{1}} = \lambda_{1} \overrightarrow{F_{1} M}$ and $\overrightarrow{P F_{2}} = \lambda_{2} \overrightarrow{F_{2} N}$, find the coordinates of point $P$ such that the slope of the line segment $MN$ is $-\frac{1}{9}$.

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