
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 a hyperbola $C$ and an ellipse $\dfrac{x^{2}}{16} + \dfrac{y^{2}}{12} = 1$ sharing common foci $F\_1$ and $F\_2$. The eccentricities of the hyperbola and the ellipse are reciprocals of each other. If the distance from a point $P$ on the right branch of the hyperbola to the right focus $F\_2$ is $4$, what is the distance from the midpoint $M$ of $PF\_2$ to the origin $O$?

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