
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 $C:\frac{{y}^{2}}{9}+\frac{{x}^{2}}{8}=1$ with the upper and lower foci $F_{1}$, $F_{2}$, and the origin $O$ at the coordinate origin.<br/>$(1)$ If point $P$ lies on the ellipse $C$ and $|PF_{1}|=|PF_{2}|$, find the cosine value of $\angle F_{1}PF_{2}$;<br/>$(2)$ If the line $l:x-y+1=0$ intersects the ellipse $C$ at points $A$ and $B$, with $M$ as the midpoint of the line segment $AB$, find the slope of the line $OM$.

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