
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 ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1(a > b > 0)$ has its left and right foci as $F_1$ and $F_2$, respectively. For any point $P$ on the ellipse $M$ and the maximum value of $\left| PF_1 \right|\left| PF_2 \right|$ is in the range of $\left[ 2c^2,3c^2 \right]$, where $c=\sqrt{a^2-b^2}$, find the range of the eccentricity $e$ of the ellipse.

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