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Assessing a person's circulation primarily involves evaluating their pulse and breathing. The pulse indicates blood flow and heart function, while breathing reflects the body's overall oxygenation and circulatory status. Checking the pulse allows you to determine if the heart is pumping effectively and whether there is adequate blood circulation to the extremities. Additionally, assessing breathing provides insights into the effectiveness of circulation, as good circulation is necessary for oxygen transport throughout the body.

When considering other options, asking the person to squeeze your hand is a responsiveness check more than a direct assessment of circulation, indicating neurological status rather than cardiovascular health. Monitoring temperature can give clues about circulation in cases of extreme conditions but is not a direct measure of blood flow. Observing for signs of swelling, while potentially relevant in certain medical situations, does not directly assess circulation's effectiveness or the current state of blood flow. Thus, checking pulse and breathing is the most appropriate method for assessing a person's circulatory status.