
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 geometric sequence $\left\{ a_n \right\}$ with a common ratio $q > 1$, and it satisfies: $a_2 + a_3 + a_4 = 28$, and $a_3 + 2$ is the arithmetic mean of $a_2$ and $a_4$.

$(1)$ Find the general formula for the sequence $\left\{ a_n \right\}$;

$(2)$ If the sequence $b_n = \log_2 a_{n+5}$, and the sum of the first $n$ terms of the sequence $\left\{ b_n \right\}$ is $S_n$, find the sum of the first $n$ terms of the sequence $\left\{ \frac{S_n}{n} \right\}$, denoted as $T_n$.

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