
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
Imagine the segment of the X-axis between the points \( x_{1} \) and \( x_{2} \) as a weightless rod. Place a weight \( p_{1} \) at the endpoint \( x_{1} \) and a weight \( p_{2} \) at the endpoint \( x_{2} \). At which point on the X-axis will the center of mass of the rod be located?

Place a weight \( p_{1} \) at the point \( \left( x_{1}, y_{1} \right) \) on the curve \( y=f(x) \) and a weight \( p_{2} \) at the point \( \left( x_{2}, y_{2} \right) \). If we imagine these points connected by a weightless rod, where will the center of mass of the rod be now? (What will be its abscissa and ordinate?)

What does Jensen's inequality mean physically? What happens if the weights \( p_{1} \) and \( p_{2} \) are equal?

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