
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
The cubic function \( y = x^3 + ax^2 + bx + c \) intersects the x-axis at points \( A, T, \) and \( B \) in that order. Tangents to the cubic curve at points \( P \) and \( Q \) are drawn from points \( A \) and \( B \) respectively, where \( P \) does not coincide with \( A \) and \( Q \) does not coincide with \( B \). The ratio of the projections of vectors \( \overrightarrow{AB} \) and \( \overrightarrow{PQ} \) onto the x-axis is:

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