
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
Observe the sequence 2, 4, 8, 16, 32, ..., it is found that starting from the second term, the ratio of each term to the previous term is a constant, this constant is $ \boxed{2}$; according to this rule, if $a_n$ (n is a positive integer) represents the nth term of this sequence, then $a_{18} = \boxed{2^{18}}$, and $a_n = \boxed{2^n}$;  
(2) If you want to find the value of $1+3+3^2+3^3+...+3^{20}$, let $S=1+3+3^2+3^3+...+3^{20}$…(1)  
Multiply both sides of equation (1) by 3, to get $ \boxed{3S=3+3^2+3^3+3^4+...+3^{21}}$…(2)  
By subtracting equation (1) from equation (2), we get $S= \boxed{\frac{3^{21}-1}{2}}$.  
(3) By using the method from specific to general, we know: If the sequence $a_1, a_2, a_3, ..., a_n$, starting from the second term, the constant ratio of each term to its previous term is q, then $a_n = \boxed{a_1q^{n-1}}$, and if this constant q ≠ 1, then $S_n = a_1 + a_2 + a_3 + ... + a_n = \boxed{\frac{a_1(q^n-1)}{q-1}}$ (expressed in an algebraic formula containing $a_1$, q, and n).  
(4) Given that the sequence satisfies (3), and $a_6 - a_4 = 24$, $a_3a_5 = 64$, find $S_8 = a_1 + a_2 + a_3 + ... + a_8$.

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