
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 $xOy$, given points $A(1,0)$ and $B(4,0)$, there are exactly two points $P$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{a^2-1}=1$ such that $\frac{PA}{PB}=\frac{1}{2}$. The set of values for the real number $a$ is ______.

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