
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
10 students are arranged in a row. Every minute, a new student is inserted in the row (which can occur in the front and in the back as well, hence  $11$  possible places) with a uniform  $\tfrac{1}{11}$  probability of each location. Then, either the frontmost or the backmost student is removed from the row (each with a  $\tfrac{1}{2}$  probability). 

Suppose you are the eighth in the line from the front. The probability that you exit the row from the front rather than the back is  $\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $100m+n$ .

*Proposed by Lewis Chen*

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