
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
Let  $n$  be a positive integer, and let  $V_n$  be the set of integer  $(2n+1)$ -tuples  $\mathbf{v}=(s_0,s_1,\cdots,s_{2n-1},s_{2n})$  for which  $s_0=s_{2n}=0$  and  $|s_j-s_{j-1}|=1$  for  $j=1,2,\cdots,2n$ . Define
\[
q(\mathbf{v})=1+\sum_{j=1}^{2n-1}3^{s_j},
\]
and let  $M(n)$  be the average of  $\frac{1}{q(\mathbf{v})}$  over all  $\mathbf{v}\in V_n$ . Evaluate  $M(2020)$ .

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