
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 ellipse $M$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ are denoted as $F_1$ and $F_2$, respectively. Let $P$ be any point on the ellipse $M$, and the range of the maximum value of $|PF_1| \cdot |PF_2|$ is $[2c^2, 3c^2]$, where $c= \sqrt{a^2-b^2}$. Then, the range of the eccentricity $e$ of the ellipse $M$ is ______.

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