
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 a fixed regular pentagon ABCDE with side 1, let M be an arbitrary point inside or on it. Let the distance from M to the closest vertex be \( r_1 \), to the next closest be \( r_2 \), and so on, such that the distances from M to the five vertices satisfy \( r_1 \leq r_2 \leq r_3 \leq r_4 \leq r_5 \). Find:
(a) the locus of M which gives \( r_3 \) the minimum possible value,
(b) the locus of M which gives \( r_3 \) the maximum possible value.

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