
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
Let $F$ be the left focus of the hyperbola $\Gamma: x^{2}-y^{2}=1$. A line passing through $F$ intersects $\Gamma$ at points $M$ and $N$. 
(1) If both $M$ and $N$ are on the left branch of the hyperbola, find the minimum area of the triangle $\triangle OMN$;
(2) Does there exist a point $P$ on the $x$-axis such that $\overrightarrow{PM} \cdot \overrightarrow{PN}$ is constant? If such a point $P$ exists, find its coordinates; if not, explain why.

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