
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 an ellipse with its center at the origin and its foci on the $x$-axis, and it passes through the point $(3,\sqrt{2})$, with an eccentricity of $\frac{\sqrt{6}}{3}$,

$(1)$ Find the equation of the ellipse;

$(2)$ $F_1$ and $F_2$ are the left and right foci of the ellipse, respectively. Let $P$ be a point on the ellipse. If $\angle F_1PF_2$ is acute, find the range of the $x$-coordinate of point $P$.

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