
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 parabola $y^{2}=2px(p > 0)$ with focus $F$ and directrix $l$. A line passing through point $F$ intersects the parabola at points $A$ and $B$. Perpendicular lines from $A$ and $B$ to the directrix $l$ meet $l$ at points $C$ and $D$ respectively. Let $O$ be the origin. The following conclusions are drawn: 

1. $\overrightarrow{AC}+\overrightarrow{CD}=\overrightarrow{BD}-\overrightarrow{BA}$;
2. There exists a real number $\lambda \in \mathbb{R}$ such that $\overrightarrow{AD}=\lambda\overrightarrow{AO}$ holds;
3. $\overrightarrow{FC}\cdot\overrightarrow{FD}=0$;
4. For any point $M$ on the directrix $l$, $\overrightarrow{AM}\cdot\overrightarrow{BM} > 0$.

Among these conclusions, the correct ones are __________. (Write down all the correct conclusion numbers)

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