
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 the sum of the first $n$ terms as $S_{n}$, $a_{3}+a_{6}=a_{5}+4$, $a_{2}$, $a_{4}$, $2a_{4}$ form a geometric sequence.<br/>$(1)$ Find the general formula for $\{a_{n}\}$ and $S_{n}$;<br/>$(2)$ Let the sequence $\left\{{\frac{{50}}{{{S_n}}}}\right\}$ have a sum of the first $n$ terms as $T_{n}$, find the number of elements in the set $\{m|T_{m}=100-a_{k}$, $k\in N_{+}$, $m\in N_{+}$, and $k\geqslant 10\}$.

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