
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 Cartesian coordinate system, the left and right foci of the ellipse $C_1$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ are $F_1$ and $F_2$, respectively, where $F_2$ is also the focus of the parabola $C_2$: $y^2 = 4x$. Point $P$ is the intersection of $C_1$ and $C_2$ in the first quadrant, and $|PF_2| = \frac{5}{3}$.
- (Ⅰ) Find the equation of the ellipse;
- (Ⅱ) A line passing through $F_2$ 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 segment $OF_2$ such that the quadrilateral with $TM$ and $TN$ as adjacent sides is a rhombus, find the range of values for $t$.

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