
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 the ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 (a > b > 0)$ has a focus at $F(1,0)$, and $O$ is the origin. The two trisection points of the minor axis of the ellipse and one focus form an equilateral triangle. A line passing through point $H(3,0)$ intersects the ellipse $C$ at points $A$ and $B$.

   (I) Find the equation of the ellipse $C$;

   (II) Let $P$ be a point on the ellipse, and it satisfies $\overrightarrow{OA}+\overrightarrow{OB}=t\overrightarrow{OP}$, when $|\overrightarrow{PA}-\overrightarrow{PB}| < \sqrt{3}$, find the range of values for $t^2$.

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