
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 sum of the coefficients of the expansion of $(2x-1)^{n}$ is less than the sum of the binomial coefficients of the expansion of $(\sqrt{x}+\frac{1}{2\sqrt[4]{x}})^{2n}$ by $255$.  
$(1)$ Find all the rational terms of $x$ in the expansion of $(\sqrt{x}+\frac{1}{2\sqrt[4]{x}})^{2n}$;  
$(2)$ If $(2x-1)^{n}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\ldots+a_{n}x^{n}$, find the value of $(a_{0}+a_{2}+a_{4})^{2}-(a_{1}+a_{3})^{2}$.

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