
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
In the Cartesian coordinate system \(xOy\), there are two fixed points \(M(0, -\frac{1}{3})\) and \(N(0, \frac{1}{3})\). For a moving point \(P\) in the plane, its projection on the \(u\)-axis is \(P_1\), and \(|\overrightarrow{MN} + \overrightarrow{MP_1}| = |\overrightarrow{NM} + \overrightarrow{NP}|\). The trajectory of point \(P\) is denoted as \(C\).
(Ⅰ) Find the equation of the trajectory \(C\) of point \(P\);
(Ⅱ) Suppose point \(F(0,1)\), \(A(2,1)\), with \(A\) as the center and \(|AF|\) as the radius, the circle centered at \(A\) is tangent to the line \(y=-1\) at point \(B\). A line with a positive slope passing through \(F\) intersects curve \(C\) in the first quadrant at point \(Q\) and intersects the circle at point \(H\). If the slopes of lines \(QH\), \(QA\), \(QB\) form an arithmetic sequence, and \(E\) is the midpoint of \(QB\), find the ratio of the areas of triangles \(QFB\) and \(QHE\).

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