
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
We have  $2022$   $1s$  written on a board in a line. We randomly choose a strictly increasing sequence from  ${1, 2, . . . , 2022}$  such that the last term is  $2022$ . If the chosen sequence is  $a_1, a_2, ..., a_k$  ( $k$  is not fixed), then at the  $i^{th}$  step, we choose the first a $_i$  numbers on the line and change the 1s to 0s and 0s to 1s. After  $k$  steps are over, we calculate the sum of the numbers on the board, say  $S$ . The expected value of  $S$  is  $\frac{a}{b}$  where  $a, b$  are relatively prime positive integers. Find  $a + b.$ 

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