
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 $\{a_{n}\}$ that satisfies $a_{n}+a_{n+2}=2a_{n+1}$, $n\in N^{*}$ and $a_{1}=1$, $a_{2}=3$.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ If the sum of the first $n$ terms of a sequence $\{b_{n}\}$ is $S_{n}$, $b_{1}=1$, and it satisfies $3S_{n}=b_{n+1}-1$, let ${c_n}=\frac{1}{{{a_n}({{{log}_2}{b_n}+3})}}$, find the sum of the first $n$ terms of the sequence $\{c_{n}\}$ denoted by $T_{n}$.

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