
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
Point \( P \) is located on the diameter of a circle with radius \( R \) between the ends of the diameter \( AB \). From point \( P \), three unit masses move along the directions of segments \( PA \), \( PB \), and \( PC \) such that \( PC \) is a semicord perpendicular to the diameter \( AB \). At what distance from \( A \) is point \( P \) if it is known that the velocities are constant and within one unit of time, the first mass reached point \( A \), the second mass reached point \( B \), and the third mass reached point \( C \)? Moreover, the total kinetic energy expended ( \(\boldsymbol{m}^{2} / 2\) ) is \( a^{2} \) units. Within what limits can the value of \( a^{2} \) vary for the condition of the problem to be satisfied?

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