
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 $A$, define the encryption algorithm $f_{i}$ on $A$: when $i$ is odd, increase the value of each odd element in $A$ by $i$ and decrease the value of each even element by $1$; when $i$ is even, increase the value of each even element in $A$ by $2i$ and decrease the value of each odd element by $2$, and denote the new sequence obtained as $f_{i}(A)$ $(i\in N^{*})$. Let sequence $B_{0}$ be: $2$, $0$, $2$, $3$, $5$, $7$, and define $B_{n}=f_{n}(B_{n-1})$ for $(n\in N^{*})$. Then $B_{2}$ is ____; the sum of all terms in $B_{2n}$ is ____.

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