
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
Siva has the following expression, which is missing operations:  $$ \frac12 \,\, \_  \,\,\frac14 \,\, \_ \,\, \frac18 \,\, \_  \,\,\frac{1}{16} \,\, \_  \,\,\frac{1}{32}. $$   For each blank, he flips a fair coin: if it comes up heads, he fills it with a plus, and if it comes up tails, he fills it with a minus. Afterwards, he computes the value of the expression. He then repeats the entire process with a new set of coinflips and operations. If the probability that the positive difference between his computed values is greater than  $\frac12$  can be expressed as  $\frac{a}{b}$  for relatively prime positive integers  $a$ ,  $b$ , then find  $a + b$ .

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