
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 $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $M:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$, $A$ is the right vertex, $B$ is the top vertex. If there is exactly one point $P$ on the line segment $AB$ such that $\overrightarrow{PF}_{1} \cdot \overrightarrow{PF}_{2} = -\frac{{c}^{2}}{2}$, then the range of the eccentricity of the ellipse $M$ is ______ (expressed in set or interval form).

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