
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 \( F \) is the right focus of the ellipse \( \frac{x^{2}}{1+a^{2}} + y^{2} = 1 \) with \( a > 0 \), and \( M(m, 0) \) and \( N(0, n) \) are moving points on the \( x \)-axis and \( y \)-axis respectively, satisfying \( \overrightarrow{MN} \cdot \overrightarrow{NF} = 0 \). Let point \( P \) satisfy \( \overrightarrow{OM}=2\overrightarrow{ON}+\overrightarrow{PO} \).

(1) Find the trajectory \( C \) of point \( P \).
(2) Through point \( F \), draw any line that intersects trajectory \( C \) at points \( A \) and \( B \). Let lines \( OA \) and \( OB \) intersect the line \( x = -a \) at points \( S \) and \( T \) respectively (\( O \) is the origin). Determine if \( \overrightarrow{FS} \cdot \overrightarrow{FT} \) is a constant value. If it is, find this constant value; if not, provide an explanation.

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