
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
We call $(\sqrt{a}+\sqrt{b})$ and $(\sqrt{a}-\sqrt{b})$ a "dual expression." The "dual expression" can be used to solve radical equations. For example, when solving the equation $\sqrt{24-x}-\sqrt{8-x}=2$, Xiaoming used the following method:<br/>Since $(\sqrt{24-x}-\sqrt{8-x})(\sqrt{24-x}+\sqrt{8-x})={(\sqrt{24-x})^2}-{(\sqrt{8-x})^2}=\left(24-x\right)-\left(8-x\right)=16$,<br/>and because $\sqrt{24-x}-\sqrt{8-x}=2$①, then $\sqrt{24-x}+\sqrt{8-x}=8$②. Adding ① and ②, we get $\sqrt{24-x}=5$,<br/>Squaring both sides of $\sqrt{24-x}=5$ gives $x=-1$, substituting $x=-1$ back into the original equation for verification, we find that $x=-1$ is a solution to the original equation.<br/>Please answer the following questions based on the above material:<br/>$(1)$ If the dual expression of $m=2-\sqrt{3}$ is $n$, then $m\times n=\_\_\_\_\_\_$ (write down the result directly).<br/>$(2)$ The solution to the equation $\sqrt{x+42}+\sqrt{x+10}=16$ is _______ (write down the result directly).<br/>$(3)$ Solve the equation: $\sqrt{4{x^2}+6x-5}+\sqrt{4{x^2}-2x-5}=4x$.

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