
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 a common difference $d$ not equal to $0$, and $\{a_7, a_3, a_1\}$ are consecutive terms of a geometric sequence $\{b_n\}$ from front to back.
$(1)$ If $a_1=4$, find the sum of the first $10$ terms of the arithmetic sequence $\{a_n\}$, denoted as $S_{10}$;
$(2)$ If the sum of the first $100$ terms of the geometric sequence $\{b_n\}$ is $T_{100}=150$, find the value of $b_2+b_4+b_6+\cdots+b_{100}$.

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