
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 parabola $C: y^2 = 2px$ ($p > 0$) with focus $F$, and point $F$ is also a focus of the hyperbola $\frac{3x^2}{5} - \frac{3y^2}{7} = 1$:
(1) Determine the equation of the parabola $C$;
(2) Draw a random line $l$ through point $F$ that intersects the curve $C$ at points $A$ and $B$.  
   (i) Find the value of $\overrightarrow{OA} \cdot \overrightarrow{OB}$;  
   (ii) From points $A$ and $B$ draw tangents to the circle $(x-2)^2 + y^2 = 1$, with the tangency points denoted as $P$ and $Q$, respectively. Let $\alpha=\angle AFP$, $\beta=\angle BFQ$, find the value of $\cos\alpha + \cos\beta$.

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