
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
In the space rectangular coordinate system, given a triangular pyramid $P-ABC$, $\overrightarrow{AB}=(2,-1,3)$, $\overrightarrow{AC}=(-2,1,0)$, $\overrightarrow{AP}=(3,-1,4)$.  
$(1)$ Find the volume of the triangular pyramid $P-ABC$.  
$(2)$ Using the method of finding trajectory equations, try to find the equation of the line passing through point $M(1,1,1)$ in the space rectangular coordinate system with direction vector $\overrightarrow{AB}=(2,-1,3)$.

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