
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
On the directrix of a parabola, choose any point \( C_1 \) such that \( C_1 \) is not on the axis of symmetry of the parabola. Draw a tangent to the parabola through \( C_1 \), and let the points of tangency be \( A_1 \) and \( B_1 \). The line \( A_1 B_1 \) intersects the directrix at \( C_2 \). Then, through \( C_2 \), draw another tangent to the parabola, with points of tangency denoted as \( A_2 \) and \( B_2 \). The line \( A_2 B_2 \) intersects the directrix at \( C_3 \). Continuing in this manner, a sequence of points \( C_1, C_2, C_3 \) is obtained. Determine the properties of this sequence of points.

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