
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 biased coin has a  $ \dfrac{6 + 2\sqrt{3}}{12} $  chance of landing heads, and a  $ \dfrac{6 - 2\sqrt{3}}{12} $  chance of landing tails. What is the probability that the number of times the coin lands heads after being flipped 100 times is a multiple of 4? The answer can be expressed as  $ \dfrac{1}{4} + \dfrac{1 + a^b}{c \cdot d^e} $  where  $ a, b, c, d, e $  are positive integers. Find the minimal possible value of  $ a + b + c + d + e $ .

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