
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 the hyperbola $C$ be defined by $\dfrac{x^2}{2} - \dfrac{y^2}{3} = 1$, with $F_1$ and $F_2$ being its left and right foci, respectively.

$(1)$ Let $O$ be the origin, and $M$ be any point on the right branch of the hyperbola $C$. Find the range of values for $\overrightarrow{OM} \cdot \overrightarrow{F_1M}$.

$(2)$ If a moving point $P$ has a constant sum of distances from the two foci $F_1$ and $F_2$ of the hyperbola $C$, and the minimum value of $\cos \angle F_1PF_2$ is $- \dfrac{1}{9}$, find the equation of the trajectory of point $P$.

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