
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 palindrome is a string that does not change when its characters are written in reverse order. Let \( S \) be a 40-digit string consisting only of 0s and 1s, chosen uniformly at random out of all such strings. Let \( E \) be the expected number of nonempty contiguous substrings of \( S \) which are palindromes. Compute the value of \( \lfloor E \rfloor \).

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