
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
In the rectangular coordinate system, the left and right foci of the ellipse (C1): $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ are F1 and F2, respectively. F2 is also the focus of the parabola (C2): $y^2 = 4x$. Point P is the intersection point of C1 and C2 in the first quadrant, and $|PF_{2}| = \frac{5}{3}$.

(I) Find the equation of the ellipse;

(II) A line passing through F2 and not perpendicular to the coordinate axes intersects the ellipse at points M and N. If there exists a fixed point T(t, 0) on the line segment OF2 such that the quadrilateral with TM and TN as adjacent sides is a rhombus, find the range of t values.

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