
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 respectively the left and right foci of the ellipse $C$: $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a > b > 0$), and $A$, $B$ are respectively the left and right vertices of the ellipse $C$, $\overrightarrow{AB}=(5+2\sqrt{6})\cdot \overrightarrow{F_2B}$, and the midpoint coordinates of $OF_2$ (where $O$ is the origin) are $(\dfrac{\sqrt{30}}{6},0)$.

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

(Ⅱ) Given that the line $y=k(x+1)$ intersects the ellipse $C$ at points $P$ and $Q$, and point $M(-\dfrac{7}{3},0)$, find the value of $\overrightarrow{MP}\cdot \overrightarrow{MQ}$.

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