
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 $xOy$, it is known that the ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ has an eccentricity $e = \dfrac{\sqrt{3}}{2}$. $F_1$ and $F_2$ are the left and right foci, respectively. A line passing through $F_1$ intersects the ellipse $C$ at points $P$ and $Q$, and the perimeter of $\triangle PQF_2$ is $8$.
1. Find the equation of the ellipse $C$.
2. Suppose a line passing through the point $M(3,0)$ intersects the ellipse $C$ at two distinct points $A$ and $B$. Let $N$ be a point on the ellipse, and it satisfies $\overrightarrow{OA} + \overrightarrow{OB} = t \overrightarrow{ON}$ ($O$ is the origin). When $|AB| < \sqrt{3}$, find the range of the real number $t$.

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