
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 points A(0, 1), B(0, -1), and C(1, 0), a moving point P satisfies the equation: $$\overrightarrow {AP}\cdot \overrightarrow {BP}=k| \overrightarrow {PC}|^{2}$$.

1. Find the trajectory equation of the moving point P and describe the curve represented by the equation.
2. When $$k=2$, find the maximum and minimum values of $$|2 \overrightarrow {AP}+ \overrightarrow {BP}|$$.

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