
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 that $a$ and $b$ are unit vectors, and $a \cdot b = -\frac{1}{2}$. If the plane vector $p$ satisfies $p \cdot a = p \cdot b = \frac{1}{2}$, calculate the magnitude of $p$.

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