
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 parabola C: y^2 = 2px (p > 0) has a directrix l and a focus F. Circle M has its center on the positive x-axis and is tangent to the y-axis. A line m passing through the origin O with an inclination angle of π/3 intersects the directrix l at point A and intersects circle M at two distinct points O and B, with |AO| = |BO| = 2. If P is a moving point on the parabola C, find the minimum value of →PM⋅→PF.

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