
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 the directrix of the parabola $y^2=2px$ ($p>0$) is tangent to the circle $(x-3)^2+y^2=225$, and one of the asymptotes of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a>0$, $b>0$) is $y=\sqrt{3}x$, and one of its foci is the focus of the parabola, find the length of the real axis of the hyperbola.

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