
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  $$ x_1x_2 \cdots x_{2022} = 1, $$    $$ (x_1 +1)(x_2 +1)\cdots (x_{2022} +1)=2, $$    $$ \text{and so on}, $$    $$ (x_1 + 2021) (x_2 + 2021) \cdots (x_{2022} + 2021) = 2^{2021}, $$  compute  $$ (x_1 +2022)(x_2 +2022) \cdots (x_{2022} +2022). $$  

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