
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 an arithmetic sequence $\{a_{n}\}$ with distinct terms and the sum of the first $4$ terms is $10$, and $a_{1}$, $a_{2}$, $a_{4}$ are the first $3$ terms of a geometric sequence $\{b_{n}\}$. <br/>$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$; <br/>$(2)$ Let ${c_n}=\frac{1}{{{a_n}{{\log}_2}{b_{n+2}}}}$, 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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