
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 function $f(2x-1)=8x^{2}-10x$.  
$(1)$ Find the analytical expression of $f(x)$;  
$(2)$ If for any $x\in [-2,2]$, $f(e^{x})\leq te^{x}-3+e^{2}$ always holds, find the range of real number $t$;  
$(3)$ Given the function $g(x)=f(x)+2x^{3}-(a+2)x^{2}+x+5$, where $0 < a < 3$, let $N$ be the maximum value of $g(x)$ on the interval $[0,1]$, and $n$ be the minimum value. Find the range of $N-n$.

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