
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 point $P$ is a point outside the plane of the parallelogram $ABCD$, and $\overset{⇀}{AB} =(2,-1,-4)$, $\overset{⇀}{AD} =(4,2,0)$, $\overset{⇀}{AP} =(-1,2,-1)$. Determine the correct conclusion(s) among the following:
$(1)$ $AP \perp AB$;
$(2)$ $AP \perp AD$;
$(3)$ $\overset{⇀}{AB}$ is the normal vector of the plane $ABCD$;
$(4)$ $\overset{⇀}{AP}$ is parallel to $\overset{⇀}{BD}$.

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