
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 function $ax^2 + px + q$ has $a > 0$ and $p > 0$, where $q$ is any real number. Find the x-coordinate of the vertex and determine the value of the function when $x = -\frac{p}{a}$. How should the function change if $a$ changes to $\frac{a}{2}$ keeping $p$ and $q$ constant?
**A)** $-\frac{p}{2a}, q$  
**B)** $-\frac{p}{a}, \frac{q}{2}$  
**C)** $-\frac{p}{4a}, q$  
**D)** $-\frac{p}{2a}, 2q$

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