
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
A bag contains 15 balls, marked with the 15 numbers $2^{0}, 2^{1}, 2^{2}, \ldots, 2^{14}$ respectively. Each ball is either red or blue, and there is at least one ball of each color. Let $a$ be the sum of the numbers on all red balls, $b$ be the sum of the numbers on all blue balls, and $d$ be the H.C.F. of $a$ and $b$. Find the greatest possible value of $d$.

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