
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
Point $P$ is any point on the ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{a^2-1}=1\left(a > 1\right)$, and $AB$ is any diameter of the circle $M$: $(x-1)^2+y^2=1$. If the maximum value of $\overrightarrow{PA} \cdot \overrightarrow{PB}$ is $15$, then $a=$____.

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