
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 the function $f\left(x\right)=\ln x$, point $A_{1}(a_{1}$,$f(a_{1}))(a_{1} \gt 0)$ is taken. The tangent line passing through $A_{1}$ intersects the $y$-axis at $(0$,$a_{2})(a_{2} \gt 0)$. Point $A_{2}(a_{2}$,$f(a_{2}))$ is then taken. The tangent line passing through $A_{2}$ intersects the $y$-axis at $(0$,$a_{3})$. This process continues until $a_{n}\leqslant 0\left(n\geqslant 3\right)$, resulting in the sequence $\{a_{n}\}$. Four conclusions are given: ①$0 \lt a_{1} \lt e$; ②For $n\geqslant 2$,$n\in N^{*}$, $a_{n}=\ln a_{n-1}-1$; ③For $n\geqslant 2$,$n\in N^{*}$, $a_{n}\leqslant a_{n-1}-2$ always holds; ④If there exists $k\in N*$ such that $a_{1}$,$a_{2}$,$\ldots $,$a_{k}$ form an arithmetic progression, then $k$ can only be $3$. The correct conclusions are represented by the numbers: ______.

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