
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 line $y=2x+2$ and the parabola $y=ax^2$ ($a > 0$) intersect at points $P$ and $Q$. When a perpendicular line is drawn from the midpoint of segment $PQ$ to the x-axis, intersecting the parabola at point $A$, if $| \overrightarrow{AP}+ \overrightarrow{AQ}|=| \overrightarrow{AP}- \overrightarrow{AQ}|$, then find the value of $a$.

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