
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 parabola $y^2=4x$ and the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 (a > 0, b > 0)$ have the same focus $F$, $O$ is the coordinate origin, points $A$ and $B$ are the intersection points of the two curves. If $(\overrightarrow{OA} + \overrightarrow{OB}) \cdot \overrightarrow{AF} = 0$, find the length of the real axis of the hyperbola.

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