
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 \(O\), \(A\), and \(B\) be three non-collinear points in a plane. The vectors are given by \(\overrightarrow{OA} = \vec{a}\) and \(\overrightarrow{OB} = \vec{b}\). Let \(P\) be an arbitrary point on the perpendicular bisector of segment \(AB\), and \(\overrightarrow{OP} = \vec{p}\). Given \(|\vec{a}| = 5\) and \(|\vec{b}| = 3\), find the value of \(\vec{p} \cdot (\vec{a} - \vec{b})\).

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