
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 that point $P(x,y)$ is a moving point on the circle $x^{2}+y^{2}=2y$,
(1) Find the range of $z=2x+y$;
(2) If $x+y+a\geqslant 0$ always holds, find the range of real numbers $a$;
(3) Find the maximum and minimum values of $x^{2}+y^{2}-16x+4y$.

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