
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

Draw on the plane the set of points whose coordinates satisfy the condition

$$
x^{2}+i-2 x+2 y i=y-1+\frac{4 y^{2}-1}{2 y-1} i
$$

Draw the sets of points that satisfy the given conditions ($10.40-10.42$):

$10.40 \ |z-2 i| \leq |z+1|$.

$10.41 \ \left\{
\begin{array}{l}
2 \leq |z-1| \leq 3 \\
0 < \operatorname{Im} z < \sqrt{5}
\end{array}
\right.
$

$10.42 \ \left\{
\begin{array}{l}
1 \leq |z+0.5 i| \leq 2, \\
0.5 < \operatorname{Re} z \leq 1
\end{array}
\right.
$

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