
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 right focus F of the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > b > 0)$ is used to drop a perpendicular line to the x-axis, which intersects the hyperbola C at two points M and N. Let A be the left vertex of the hyperbola, and $\angle MAN = \theta$. The eccentricity of the hyperbola C is denoted as $f(\theta)$. Then find the value of $f\left(\frac{2\pi}{3}\right) - f\left(\frac{\pi}{3}\right)$.

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