
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 an ellipse $\frac{x^{2}}{p^{2}} + \frac{y^{2}}{3} = 1$, the left focus point is on the directrix of the parabola $C: y^2 = 2px$ ($p > 0$). Let $F$ be the focus of the parabola.
(I) Find the equation of the parabola $C$.
(II) If line $l$ passes through point $F$ and intersects the parabola at two distinct points $A$ and $B$, and intersects the y-axis at point $M$, and $\overrightarrow{MA} = a \overrightarrow{AF}$, $\overrightarrow{MB} = b \overrightarrow{BF}$, then for any line $l$, is $a + b$ a constant value? If so, what is it? If not, please explain why.

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