
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 $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, $a_{1}=1$, $a_{2}=2$, $\left(2n+3\right)S_{n+1}=\left(n+2\right)S_{n}+\left(n+1\right)S_{n+2}$ for $n\in N^{*}$.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ Given that the sum of the first $n$ terms of the sequence $\left\{\log_2\frac{a_{n+1}}{a_n}\right\}$ is $T_{n}$, $c_{n}=[T_{n}]$ (where the function $\left[x\right]$ represents the integer not exceeding $x$, such as $\left[2.1\right]=2$), find the sum of the first $100$ terms of the sequence $\{c_{n}\}$, denoted as $M_{100}$.

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