
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 sequence of vectors $\{\vec{a_n}\}$: $\vec{a_1}=(x_1,y_1)$, $\vec{a_2}=(x_2,y_2)$, ..., $\vec{a_n}=(x_n,y_n)$, if from the second term, the difference between each term and the previous term is equal to the same vector, then this sequence of vectors is called an arithmetic vector sequence. It is known that the arithmetic vector sequence $\{\vec{a_n}\}$ satisfies $\vec{a_1}=(-20,13)$, $\vec{a_3}=(-18,15)$, find the serial number $n=$______ of the vector with the smallest modulus in the sequence of vectors $\{\vec{a_n}\}$.

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