
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 a hyperbola with the equation $\frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1$ (where $a > 0$, $b > 0$), a point $P(x_0, y_0)$ on the hyperbola has a tangent line with the equation $\frac {x_{0}x}{a^{2}} - \frac {y_{0}y}{b^{2}} = 1$. If a point $P(x_0, y_0)$ on the hyperbola ($a \leq x_0 \leq 2a$) has a tangent line that passes through the point $(0, b)$, and the slope of this tangent line is $-2$, find the eccentricity of the hyperbola.

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