
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
The left and right foci of the hyperbola $\dfrac {x^{2}}{a^{2}}- \dfrac {y^{2}}{b^{2}}=1(a > 0,b > 0)$ are $F\_1$ and $F\_2$, respectively. The left and right vertices are $A\_1$ and $A\_2$, respectively. A line passing through focus $F\_2$ and perpendicular to the $x$-axis intersects the hyperbola at point $P$. If $| \overrightarrow{PA\_1}|$ is the geometric mean of $| \overrightarrow{F\_1F\_2}|$ and $| \overrightarrow{A\_1F\_2}|$, then the eccentricity of the hyperbola is _______.

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