
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 game of backgammon has a "doubling" cube, which is like a standard 6-faced die except that its faces are inscribed with the numbers 2, 4, 8, 16, 32, and 64, respectively.  After rolling the doubling cube four times at random, we let  $a$  be the value of the first roll,  $b$  be the value of the second roll,  $c$  be the value of the third roll, and  $d$  be the value of the fourth roll.  What is the probability that  $\frac{a + b}{c + d}$  is the average of  $\frac{a}{c}$  and  $\frac{b}{d}$  ?

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