
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 non-zero common difference $d$, and $a\_7$, $a\_3$, $a\_1$ are three consecutive terms of a geometric sequence ${\_{b\_n}}$.

(1) If $a\_1=4$, find the sum of the first 10 terms of the sequence ${\_{a\_n}}$, denoted as $S_{10}$;
(2) If the sum of the first 100 terms of the sequence ${\_{b\_n}}$, denoted as $T_{100}=150$, find the value of $b\_2+b\_4+b\_6+...+b_{100}$.

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