
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
Find the smallest positive integer value of  $N$  such that field  $K=\mathbb{Q}(\sqrt{N},\ \sqrt{i+1})$ , where  $i=\sqrt{-1}$ , is Galois extension on  $\mathbb{Q}$ , then find the Galois group  $Gal(K/\mathbb{Q}).$ 

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