
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
Let point $P$ be a point on the left branch of the hyperbola with foci $F_1$ and $F_2$ given by $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a > 0, b > 0$), and it satisfies $\overrightarrow{PF_1} \cdot \overrightarrow{PF_2} = 0$, $\tan \angle PF_2F_1 = \dfrac{2}{3}$. Determine the eccentricity of this hyperbola.

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