
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 the center M of the circle x^2 + y^2 + 4x + 2y + 1 = 0 and the focus F of the parabola C: y^2 = 2px are symmetric about the line 3x + y + 2 = 0, O is the origin, and the line l passes through the point P(2, 0) intersecting the parabola C at points A and B, if |BF| = 3/2 and |AP| = t|BP|, find the value of t.

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