
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 the sequence {a<sub>n</sub>} is an arithmetic sequence with the sum of the first n terms denoted as S<sub>n</sub>, and the common difference is not 0, if a<sub>4</sub>, a<sub>5</sub>, and a<sub>7</sub> form a geometric sequence, determine the value of a<sub>7</sub> given that S<sub>11</sub> = 66.

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