
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 the hyperbola $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ ($b>0$) with foci $F_{1}$ and $F_{2}$, a circle $O$ passing through $F$ (where $O$ is the origin) intersects the left branch of hyperbola $C$ at points $M$ and $N$, forming an equilateral triangle $\triangle OMN$.
- $(1)$ Find the standard equation and asymptote equation of hyperbola $C$.
- $(2)$ If point $P$ lies on the right branch of hyperbola $C$, and rays $PF_{1}$ and $PF_{2}$ intersect hyperbola $C$ at points $A$ and $B$ respectively, investigate whether $\frac{|PF_{1}|}{|AF_{1}|}-\frac{|PF_{2}|}{|BF_{2}|}$ is a constant value. If it is, find the value. If not, explain why.

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