
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) Given a sequence $\{a_n\}$ that satisfies $a_1a_2…a_n=n+1$, find $a_3=$

(2) Let $\overrightarrow{e_1}, \overrightarrow{e_2}$ be unit vectors, where $\overrightarrow{a}=2\overrightarrow{e_1}+\overrightarrow{e_2}, \overrightarrow{b}=\overrightarrow{e_2}$, and $\overrightarrow{a} \cdot \overrightarrow{b}=2$, find $|\overrightarrow{a}|=$

(3) For an arithmetic sequence with the first term $-24$, starting from the 9th term, the terms become positive. Find the range of the common difference $d$.

(4) In the sequence $\{a_n\}$, if the point $(n,a_n)$ lies on a fixed line $l$ passing through the point $(5,3)$, find the sum of the first 9 terms of the sequence, $S_9=$

(5) In rectangle $ABCD$, where $AB=2AD=2$, if $P$ is a moving point on $DC$, find the minimum value of $\overrightarrow{PA} \cdot \overrightarrow{PB}-\overrightarrow{PA} \cdot \overrightarrow{BC}=$

(6) In $\triangle ABC$, where angle $A$ is the largest and angle $C$ is the smallest, and $A=2C$, $a+c=2b$, find the ratio $a:b:c=$

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