
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 the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with left and right foci $F\_1$ and $F\_2$, a point $P$ moves on the ellipse. The maximum value of $| \overrightarrow{PF}\_{1}|×| \overrightarrow{PF\_{2}}|$ is $m$, and the minimum value of $ \overrightarrow{PF\_{1}}⋅ \overrightarrow{PF\_{2}}$ is $n$. If $m \geqslant 2n$, then the range of the eccentricity of the ellipse is _______ .

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