
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 natural number nonzero n so that it exists in real numbers  $x_1, x_2,..., x_n$  which simultaneously check  the conditions:
1)  $x_i \in [1/2 , 2]$  ,  $i = 1, 2,... , n$  
2)   $x_1+x_2+...+x_n \ge \frac{7n}{6}$ 3)  $\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}\ge \frac{4n}{3}$ 

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