
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 that $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $E: \dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1 \left(a > 0, b > 0\right)$, $P$ is a point on the right branch of the hyperbola $E$, $M$ is the midpoint of the line segment $F_{1}P$, and $O$ is the origin. If the perimeter of $\triangle OF_{1}M$ is $c + 3a$ (where $c$ is half of the hyperbola's focal length) and $\angle F_{1}MO = \dfrac{\pi}{3}$, find the equation of the asymptotes of the hyperbola $E$.

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