
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 vectors $\overrightarrow m=(a,b,0),\overrightarrow n=(c,d,1)$ where $a^{2}+b^{2}=c^{2}+d^{2}=1$, the following propositions are given:<br/>$(1)$ The angle between vector $\overrightarrow n$ and the positive $z$-axis is always a constant value (independent of $c$ and $d$);<br/>$(2)$ The maximum value of $\overrightarrow m \cdot \overrightarrow n$ is $\sqrt{2}$;<br/>$(3)$ The maximum value of $\left\langle{\overrightarrow m,\overrightarrow n}\right\rangle$ (the angle between $\overrightarrow m \cdot \overrightarrow n$) is $\frac{3\pi}{4}$;<br/>$(4)$ If we define $\overrightarrow u \times \overrightarrow v=|\overrightarrow u| \cdot |\overrightarrow v| \sin\left\langle{\overrightarrow u,\overrightarrow v}\right\rangle$, then the maximum value of $|\overrightarrow m \times \overrightarrow n|$ is $\sqrt{2}$.<br/>Which propositions are correct? (Write down the numbers of all correct propositions)

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