
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 $F\_1$ and $F\_2$ are the upper and lower foci of the ellipse $C\_1$: $\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1 (a > b > 0)$, respectively, where $F\_1$ is also the focus of the parabola $C\_2$: $x^2 = 4y$. Point $M$ is the intersection point of $C\_1$ and $C\_2$ in the second quadrant, and $|MF\_1| = \frac{5}{3}$.

(I) Find the equation of the ellipse;
(II) Given point $P(1, 3)$ and circle $O$: $x^2 + y^2 = b^2$, a moving straight line $l$ passes through point $P$ and intersects the circle $O$ at two distinct points $A$ and $B$. Take a point $Q$ on the line segment $AB$ such that $\overrightarrow{AP} = -λ\overrightarrow{PB}$ and $\overrightarrow{AQ} = λ\overrightarrow{QB} (λ ≠ 0 \text{ and } λ ≠ ±1)$. Investigate whether there exists a straight line such that point $Q$ always lies on it, and if so, find its equation.

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