
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 $a_{n+m}=a_{n}+a_{m}$ for any $m$, $n\in N^{*}$, and a geometric sequence $\{b_{n}\}$ where $b_{1}=a_{1}$, $b_{2}-a_{2}=0$, $b_{3}-a_{3}=1$.<br/>$(1)$ Find the general term formula for sequences $\{a_{n}\}$ and $\{b_{n}\}$;<br/>$(2)$ Let $c_{n}=\frac{a_{n}}{b_{n}}$, find the sum of the first $n$ terms of the sequence $\{c_{n}\}$ denoted as $T_{n}$.

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