
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: Circle $C$ passes through point $D(0,1)$, $E(-2,1)$, $F(-1,\sqrt{2})$, $P$ is any point on the line $l_{1}: y=x-2$, and the line $l_{2}: y=x+1$ intersects circle $C$ at points $A$ and $B$. <br/>$(Ⅰ)$ Find the equation of circle $C$;<br/>$(Ⅱ)$ Find the minimum value of $|PA|^{2}+|PB|^{2}$.

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