
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 hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1(a＞0,b＞0)$, where $O$ is the origin, $F_{1}$ and $F_{2}$ are the left and right foci, respectively. If there exists a point $P$ on the left branch such that the midpoint $M$ of $F_{2}P$ satisfies $|OM|=\frac{1}{5}c$, then the range of eccentricity $e$ of the hyperbola is ______.

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