
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
The sum of the first n terms of the sequence $\{a_n\}$ is $S_n$. If the terms of the sequence $\{a_n\}$ are arranged according to the following pattern: $\frac{1}{2}$, $\frac{1}{3}$, $\frac{2}{3}$, $\frac{1}{4}$, $\frac{2}{4}$, $\frac{3}{4}$, $\frac{1}{5}$, $\frac{2}{5}$, $\frac{3}{5}$, $\frac{4}{5}$…, $\frac{1}{n}$, $\frac{2}{n}$,…, $\frac{n-1}{n}$,…, the following operations and conclusions are obtained:  
① $a_{24} = \frac{3}{8}$;  
② The sequence $a_1$, $a_2+a_3$, $a_4+a_5+a_6$, $a_7+a_8+a_9+a_{10}$,… is a geometric sequence;  
③ The sum of the first n terms of the sequence $a_1$, $a_2+a_3$, $a_4+a_5+a_6$, $a_7+a_8+a_9+a_{10}$,… is $T_n = \frac{n^2+n}{4}$;  
④ If there exists a positive integer k such that $S_k < 10$, $S_{k+1} \geq 10$, then $a_k = \frac{5}{7}$.  
Among these conclusions, the correct ones are \_\_\_\_\_\_. (Fill in the numbers of all correct conclusions)

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