
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
Read the material and solve the problem. Xiao Ming encountered the following question: Calculate $\left(2+1\right)(2^{2}+1)(2^{4}+1)(2^{8}+1)$. After observation, Xiao Ming found that if the original expression is appropriately transformed, a special structure can appear, which can then be solved using the square of a binomial formula. The specific solution is as follows: 

$(2+1)(2^{2}+1)(2^{4}+1)(2^{8}+1)$  
$=\left(2-1\right)\left(2+1\right)(2^{2}+1)(2^{4}+1)(2^{8}+1)$  
$=(2^{2}-1)(2^{2}+1)(2^{4}+1)(2^{8}+1)$  
$=(2^{4}-1)(2^{4}+1)(2^{8}+1)$  
$=(2^{4}-1)(2^{8}+1)$  
$=2^{16}-1$.  

Please try to solve the following problems based on Xiao Ming's method:  
$(1)\left(2+1\right)(2^{2}+1)(2^{4}+1)(2^{8}+1)(2^{16}+1)$;  
$(2)\left(3+1\right)(3^{2}+1)(3^{4}+1)(3^{8}+1)(3^{16}+1)$.

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