
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 $\frac{x^{2}}{1+a^{2}} + y^{2} = 1$ for $a > 0$, $M(m, 0)$ and $N(0, n)$ are points on the $x$-axis and $y$-axis respectively, and they satisfy $\overrightarrow{M N} \cdot \overrightarrow{N F} = 0$. Let point $P$ satisfy $\overrightarrow{O M} = 2 \overrightarrow{O N} + \overrightarrow{P O}$.

(1) Find the locus $C$ of point $P$.

(2) For any line passing through point $F$ that intersects the locus $C$ at points $A$ and $B$, let lines $O A$ and $O B$ intersect the line $x = -a$ at points $S$ and $T$ respectively (with $O$ being the origin). Determine whether $\overrightarrow{F S} \cdot \overrightarrow{F T}$ is a constant value. If it is, find this constant; if not, provide an explanation.

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