
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 curve $C$ defined by the equation $y^{2}=2x+a$ with $a > 0$, let $P_{n}(n, \sqrt{2n+a})$ where $n\in\mathbb{N}$ be a point on this curve. The slope of the tangent line $l_{n}$ at point $P_n$ is $k_{n}$. The line $l_{n}$ intersects the $x$-axis and the $y$-axis at points $A_{n}(x_{n},0)$ and $B_{n}(0,y_{n})$, respectively, and it is given that $|x_{0}|=|y_{0}|$. Consider the following statements:

1. $a=1$;
2. For $n \in \mathbb{N}^*$, the minimum value of $y_{n}$ is $\dfrac{2 \sqrt{3}}{3}$;
3. For $n \in \mathbb{N}^*$, $k_{n} > \sqrt{2}\sin \dfrac{1}{\sqrt{2n+1}}$;
4. For $n \in \mathbb{N}^*$, if we denote the sum of the first $n$ terms of the sequence $\{k_{n}\}$ as $S_{n}$, then $S_{n} < \sqrt{2}(\sqrt{n+1}-1)$.

Identify all the correct conclusions from the above list.

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