
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
Ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \left(a > b > 0\right)$ has left and right foci $F$ and $F'$ respectively, with eccentricity $\frac{2}{3}$. $A$ is the left vertex of ellipse $C$, and $\overrightarrow{AF} \cdot \overrightarrow{AF'} = 5$. A line passing through the origin intersects ellipse $C$ at points $M$ and $N$. Find the range of $\frac{1}{|FM|} + \frac{4}{|FN|}$.

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