
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 the hyperbola $C: x^{2} - \frac{y^{2}}{3} = 1$ with left and right foci $F_1$ and $F_2$ respectively, a point $A$ lies on the hyperbola, and point $M$ has coordinates $(\frac{2}{3}, 0)$. The distances from $M$ to lines $AF_1$ and $AF_2$ are equal. Find $|AF_1|$.

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