
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 a line $e$ and points $A$ and $B$ on the same side of it at different distances. András claims that the following procedure prepares for identifying the centers of circles passing through $A$ and $B$ and tangential to $e$.

I. Point $C$ is where the line $AB$ intersects $e$.

II. Point $A^{\prime}$ is the reflection of $A$ over $e$.

III. The perpendicular bisector of segment $A^{\prime}B$ intersects the perpendicular from $C$ to $e$ at point $D$.

IV. The circle centered at $D$ with radius $D A^{\prime}$ intersects $e$ at points $T_{1}$ and $T_{2}$.

How can these steps be concluded to identify the said centers?

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