
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
(1) If the real number sequence $1$, $a$, $b$, $c$, $4$ is a geometric sequence, then the value of $b$ is $\_\_\_\_\_\_$.

(2) Given that $\{a\_n\}$ is an arithmetic sequence with common difference $d < 0$, and the sum of its first $n$ terms is denoted by $S_n$. If $S_{16} > 0$ and $S_{17} < 0$, then the value of $n$ when $S_n$ is at its maximum is $\_\_\_\_\_\_$.

(3) Given a geometric sequence $\{a\_n\}$ with positive terms and $a_2a_3=16$, the sum of the first four terms of the sequence $\{\log\_2 a\_n\}$ is $\_\_\_\_\_\_$.

(4) A municipal environmental protection bureau proposed two investment plans: Plan 1 is to invest 50 million yuan at once; Plan 2 is to invest 5 million yuan in the first year, and increase the investment by 10 million yuan each year. After how many years will the total investment of Plan 2 be no less than that of Plan 1?

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