Let \(a\in[0,1]\). The number of non-negative continuous functions \(f\) defined on \([0,1]\) which satisfy all of the following three conditions is strictly less than

**Conditions**

- \(\displaystyle \int_{0}^{1}f(x)\, dx=1\)
- \(\displaystyle \int_{0}^{1}xf(x)\, dx=a\)
- \(\displaystyle \int_{0}^{1}x^2f(x)\, dx=a^2\)

**Your options**

- A. 1
- B. 3
- C. 5
- D. 7

Write the answer as a 4-digit string of 0s and 1s, 1 for correct option, 0 for incorrect. If your answer is 5, then it is less than 6 and 8. If your answer is C and D, then you should write 0011 as the answer. Neither option may be correct, in which case 0000 would be the answer.

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