
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
The common difference $d$ of the arithmetic sequence $\{a_n\}$ is not equal to $0$. $a_3$ is the geometric mean of $a_2$ and $a_5$. It is known that the sequence $a_2$, $a_4$, $a_{k_1}$, $a_{k_2}$, ..., $a_{k_n}$, ... forms a geometric sequence. The sum of the first $n$ terms of the sequence $\{k_n\}$ is denoted as $T_n$. Then, $2T_n+9=\boxed{\text{?}}$.

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