
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
A perpendicular line is drawn from the right focus F of the hyperbola C: x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0) to the x-axis, intersecting the hyperbola C at points M and N. Let A be the left vertex, and ∠MAN = θ. The eccentricity of the hyperbola C is denoted as f(θ). Recall that the eccentricity of a hyperbola is given by e = √(1 + (b/a)^2). Determine f(2π/3) - f(π/3).

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