
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 an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>0,b>0)$, the line $l:x-\sqrt{2}y+\sqrt{2}=0$ passes through the left vertex and the top vertex of the ellipse.  
$(1)$ Find the standard equation of the ellipse $C$;  
$(2)$ Determine whether there exists a point $P$ on the line $l$, such that the two tangent lines from point $P$ to ellipse $C$ intersect at points $A$ and $B$, and point $P$ lies on a circle with diameter $AB$. If such a point exists, find the coordinates of point $P$; if not, explain the reason.

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