
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
Let set $A$ be the solution set of the equation $2x^2+x+p=0$, and set $B$ be the solution set of the equation $2x^2+qx+2=0$. If $A \cap B = \left\{ \frac{1}{2} \right\}$, find the sets $A$, $B$, and $A \cup B$.

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