
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
Circle $O: x^{2}+y^{2}=4$ intersects the negative and positive $x$-axes at points $A$ and $B$ respectively. $MN$ is a chord of the circle perpendicular to the $x$-axis and not a diameter. Line $AM$ intersects line $BN$ at point $P$, and the locus of moving point $P$ is denoted as $E$.
$(1)$ Find the equation of the locus $E$;
$(2)$ In a Cartesian coordinate system, a line determined by an angle is called a directed line. Is there a directed line $l$ that does not pass through point $A$, such that when line $l$ intersects locus $E$ at points $C$ and $D$, $AC\perp AD$? If it exists, find a directional vector of line $l$; if not, explain the reason.

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