
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 $C\_1$ are $F\_1(-1,0)$ and $F\_2(1,0)$, which are the vertices of hyperbola $C\_2$, and one of the intersection points of ellipse $C\_1$ and hyperbola $C\_2$ is $M\left( \frac{2 \sqrt{3}}{3}, \frac{ \sqrt{3}}{3}\right)$.

(1) Find the equations of ellipse $C\_1$ and hyperbola $C\_2$;

(2) If point $P$ is a moving point on the right branch of the hyperbola, and point $Q$ is a moving point on the $y$-axis such that ${F}\_{1}P⊥{F}\_{1}Q $, determine whether line $PQ$ passes through a fixed point. If it does, find the coordinates of the fixed point. If it does not, explain the reason.

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