
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 ellipse $M$ shares the same foci with ellipse $N: \frac{x^2}{9} + \frac{y^2}{5} = 1$, and ellipse $M$ passes through the point $(0,2)$.

$(1)$ Find the length of the major axis of $M$;

$(2)$ Let line $y = x + 2$ intersect $M$ at points $A$ and $B$ (with $A$ to the right of $B$), and let $O$ be the origin. Find $\overrightarrow{OA} \cdot \overrightarrow{OB}$.

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