
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 that the foci of ellipse $C_{1}$ and hyperbola $C_{2}$ are the same, denoted as $F_{1}$ and $F_{2}$ respectively, and the eccentricities of the ellipse and hyperbola are $e_{1}$ and $e_{2}$ respectively, let point $P$ be a common point of $C_{1}$ and $C_{2}$ in the first quadrant, satisfying $|PF_{1}|=k|PF_{2}|$. Find the value of $k$ when $\frac{e_{1}}{e_{2}}=\frac{1}{k-1}$.

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