
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
The hyperbola $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\left(a \gt 0,b \gt 0\right)$ has its left vertex at $A$, right focus at $F$, and moving point $B$ on $C$. When $BF\bot AF$, $|AF|=2|BF|$, and the area of $\triangle ABF$ is $\frac{25}{4}$.<br/>$(1)$ Find the equation of the hyperbola $C$;<br/>$(2)$ If point $B$ is in the first quadrant and $\angle BFA=3\angle BAF$, find the abscissa of point $B$.

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