
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 that the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, and $a_{1}=1$, $\frac{{2{S_n}}}{a_n}=n+1$.  
$(1)$ Find the general formula for $\{a_{n}\}$;  
$(2)$ Let the sum of the first $n$ terms of the sequence $\left\{{\log_2\frac{{a_{n+1}}}{{a_n}}}\right\}$ be $T_{n}$, find the number of elements in the set $\left\{{k|{{T_k}≤10,k∈{{N}^*}}}\right\}$.

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