
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
The following  $100$  numbers are written on the board:   $$ 2^1 - 1, 2^2 - 1, 2^3 - 1, \dots, 2^{100} - 1. $$  Alice chooses two numbers  $a,b,$  erases them and writes the number  $\dfrac{ab - 1}{a+b+2}$  on the board. She keeps doing this until a single number remains on the board.

If the sum of all possible numbers she can end up with is  $\dfrac{p}{q}$  where  $p, q$  are coprime, then what
is the value of  $\log_{2}(p+q)$ ?

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