
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 real numbers \( a, b, c, d, e \) that satisfy \( 3a + 2b - c + 4d + \sqrt{133}e = \sqrt{133} \) and \( 2a^2 + 3b^2 + 3c^2 + d^2 + 6e^2 = 60 \), determine the conditions under which \( e \) reaches its extremum values and find the maximum and minimum values of \( e \).

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