
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
There are  $n$  people in line, counting  $1,2,\cdots, n$  from left to right, those who count odd numbers quit the line, the remaining people press 1,2 from right to left, and count off again, those who count odd numbers quit the line, and then the remaining people count off again from left to right $\cdots$  Keep doing that until only one person is in the line.   $f(n)$  is the number of the last person left at the first count. Find the expression for  $f(n)$  and find the value of  $f(2022)$ 

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