
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 that the line $l_{1}$: $x+y-2=0$ intersects with the line $l_{2}$: $x-2y+1=0$ at point $P$, and line $l_{1}$ intersects with the $x$-axis at point $P_{1}$. A perpendicular line to the $x$-axis passing through point $P_{1}$ intersects with line $l_{2}$ at point $Q_{1}$. A perpendicular line to the $y$-axis passing through point $Q_{1}$ intersects with line $l_{1}$ at point $P_{2}$. This process continues to obtain a series of points $P_{1}$, $Q_{1}$, $P_{2}$, $Q_{2}$, ..., $P_{2}$, $Q_{2}$, ... Denote the abscissa of point $P_n$ ($n \in N^*$) as a sequence $\{x_{n}\}$. Four conclusions are given as follows: <br/>① Point $Q_2(\frac{1}{2},\frac{3}{4})$; <br/>② The sequence $\{x_{2n}\}$ is monotonically decreasing; <br/>③ $|PP_n|^2=2×(\frac{1}{4})^{n-1}$; <br/>④ The sum of the first $n$ terms of the sequence $\{x_{n}\}$, denoted as $S_{n}$, satisfies: $2S_{n+1}+S_{n}=4n+3$. Determine the correct conclusions by filling in the blank with the corresponding numbers.

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