
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 an arithmetic sequence {a<sub>n</sub>} with a non-zero common difference d, a<sub>3</sub> is the geometric mean of a<sub>2</sub> and a<sub>5</sub>. It is known that the sequence a<sub>2</sub>, a<sub>4</sub>, a$$_{k_{1}}$$, a$$_{k_{2}}$$, …, a$$_{k_{n}}$$, … forms a geometric sequence. The sum of the first n terms of the sequence {k<sub>n</sub>} is denoted as T<sub>n</sub>. Find the value of 2T<sub>n</sub> + 9.

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