
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 $C$ with its center at the origin and its foci on the $x$-axis, the distance between the foci is $2$, and the length of the major axis is $\sqrt{2}$ times the length of the minor axis.  
$(1)$ Find the standard equation of the ellipse $C$;  
$(2)$ Let $P(2,0)$, and a line $l$ passing through the left focus $F$ of the ellipse $C$ intersects $C$ at points $A$ and $B$. If for any line $l$ satisfying the condition, the inequality $\overrightarrow{PA} \cdot \overrightarrow{PB} \leqslant \lambda (\lambda \in \mathbb{R})$ always holds, find the minimum value of $\lambda$.

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