
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 the hyperbola $C$: $\dfrac {x^{2}}{a^{2}} - \dfrac {y^{2}}{b^{2}} = 1$ ($a > 0, b > 0$) have its left and right foci as $F_{1}$ and $F_{2}$, respectively. Suppose there exists a point $P$ on the right branch of curve $C$ such that the radius of the inscribed circle of $\triangle PF_{1}F_{2}$ is $a$, with the center of the circle denoted as $M$. Also, the centroid of $\triangle PF_{1}F_{2}$ is $G$, satisfying $MG$ parallel to the $x$-axis. Calculate the eccentricity of the hyperbola $C$.

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