
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} = 4x\\) with focus \\(F\\), a line \\(l\\) passing through \\(F\\) intersects \\(C\\) at points \\(A\\) and \\(B\\), \\(M\\) is the midpoint of segment \\(AB\\), and \\(O\\) is the origin. The extensions of \\(AO\\) and \\(BO\\) intersect the line \\(x = -4\\) at points \\(P\\) and \\(Q\\), respectively.

\\((\\)I\\()\\) Find the equation of the trajectory of the moving point \\(M\\);

\\((\\)II\\()\\) Connect \\(OM\\), and find the ratio of the area of \\(\\triangle OPQ\\) to the area of \\(\\triangle BOM\\).

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