
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
In the rectangular coordinate system on a plane, there are points $A(0,1)$, $B(0,-1)$, $C(1,0)$, and point $P$ that satisfies $\overrightarrow{AP}\cdot \overrightarrow{BP}=k|\overrightarrow{PC}|^{2}$.
(1) If $k=2$, find the trajectory equation of point $P$;
(2) When $k=0$, if $|\lambda \overrightarrow{AP}+ \overrightarrow{BP}|_{\text{max}}=4$, find the value of the real number $\lambda$.

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