
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 a point of intersection of an ellipse $C\_1$ and a hyperbola $C\_2$, both sharing common foci $F\_1$ and $F\_2$. If $PF\_1 \perp PF\_2$, the eccentricity of the ellipse $C\_1$ is $e\_1$, and the eccentricity of the hyperbola $C\_2$ is $e\_2$. Given that $3e\_1 = e\_2$, find the value of $e\_1$.

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