
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 sequence $\left\{ a_n \right\}$ is a geometric sequence with a common ratio of $2$, and $a_2, a_3+1, a_4$ are in arithmetic sequence in order.  
$(1)$ Find the general formula for the sequence $\left\{ a_n \right\}$.  
$(2)$ Let $b_n=n \cdot a_n + \log_2 a_{n+1}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$.

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