
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 ${(a-2x)}^{8}={a}_{0}+{a}_{1}(x-1)+{a}_{2}{(x-1)}^{2}+⋯+{a}_{8}{(x-1)}^{8}$, where $a$ is a real number and $a \gt 0$, and the coefficient of $x^{2}$ is $81648$. Find the value of the real number $a$. Calculate:  
$(1)$ the value of $a$;  
$(2)$  
$(i)$ $(a_{0}+a_{2}+a_{4}+a_{6}+a_{8})\cdot (a_{1}+a_{3}+a_{5}+a_{7})$;  
$(ii)$ $|a_{0}|+|a_{1}|+|a_{2}|+\cdots +|a_{8}|$ (express the result in terms of powers).

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