
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 the parabola $x^{2}=4py$ $(p > 0)$ with focus $F$, the line $y=x+2$ intersects the parabola at points $A$ and $B$. $M$ is the midpoint of segment $AB$. A vertical line is drawn through $M$ that intersects the $x$-axis at point $N$. If $\overrightarrow{AF} \cdot \overrightarrow{BF} + (\overrightarrow{AF} + \overrightarrow{BF}) \cdot \overrightarrow{FN} = -1 - 5p^{2}$, find the value of $p$.

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