
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
Find the general term formula for the following sequences:

1. In the sequence ${a_n}$, if $a_1=2$ and $a_{n+1}=a_n+\ln (1+\frac{1}{n})$, then $a_n=$ \_\_\_\_\_\_;
2. In the sequence ${a_n}$, if $a_1=5$ and $a_{n+1}=2a_n+2^{n+1}-1$, then $a_n=$ \_\_\_\_\_\_;
3. If $a_n=2a_n+4^{n}+2$, find the general term formula for the sequence;
4. If $a_1=1$, $(n+1)a_{n+1}^2-na_n^2+a_{n+1}a_n=0 (n\in\mathbb{N}^*$ and $a_n > 0)$, find the general term $(a_n)$ for the sequence;
5. If $a_1=1$, $na_n=a_1+2a_2+3a_3+\cdots+(n-1)a_{n-1} (n\geqslant 2, n\in\mathbb{N}^*)$, find the general term $(a_n)$ for the sequence;
6. If $a_1=1$, $a_{n+1}=\frac{a_n}{-7a_n-6}$, find the general term $(a_n)$ for the sequence;
7. If $a_1=1$, and $a_{n+1}=a_n^2+2a_n$, find the general term $(a_n)$ for the sequence.

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