Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the maximum possible value of \[\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007},\] where, for $1\leq i\leq 2007$, $x_i$ is a nonnegative real number, and \[x_1+x_2+x_3+\cdots+x_{2007}=\pi.\] Find the value of $a+b$.
To find the maximum possible value of \(\sin^2 x_1 + \sin^2 x_2 + \sin^2 x_3 + \cdots + \sin^2 x_{2007}\) given that \(x_1 + x_2 + x_3 + \cdots + x_{2007} = \pi\), we will use the properties of the sine function and some optimization techniques.

1. **Initial Assumptions and Simplifications**:
   - We are given that \(x_i \geq 0\) and \(x_1 + x_2 + \cdots + x_{2007} = \pi\).
   - If any \(x_i \geq \frac{\pi}{2}\), we can replace \(x_i\) with \(\pi - x_i\) without changing \(\sin^2 x_i\) but reducing the sum of the angles, allowing us to increase another angle and thus potentially increase the sum of \(\sin^2 x_i\). Therefore, we can assume \(x_i \in [0, \frac{\pi}{2}]\) for all \(i\).

2. **Using Lemmas to Optimize the Sum**:
   - **Lemma 1**: If \(x + y \leq \frac{\pi}{2}\) and \(x \leq y\), then \(\sin^2 x + \sin^2 y \leq \sin^2 (x - \epsilon) + \sin^2 (y + \epsilon)\) for any \(0 \leq \epsilon \leq x\).
   - **Lemma 2**: If \(x + y \geq \frac{\pi}{2}\) and \(x \leq y\), then \(\sin^2 x + \sin^2 y \leq \sin^2 (x + \epsilon) + \sin^2 (y - \epsilon)\) for any \(0 \leq \epsilon \leq \frac{y - x}{2}\).

3. **Applying Lemmas**:
   - If \(x_i \in (0, \frac{\pi}{4})\) and there exists \(x_j \geq \frac{\pi}{2} - x_i\), we can apply Lemma 2 to send both \(x_i\) and \(x_j\) to \(\frac{x_i + x_j}{2} \geq \frac{\pi}{4}\).
   - Otherwise, we can apply Lemma 1 to send \(x_i \to 0\) and \(x_j \to x_i + x_j \leq \frac{\pi}{2}\).

4. **Simplifying the Problem**:
   - By applying the lemmas, we can reduce the problem to checking when \(x_i \in \{0\} \cup [\frac{\pi}{4}, \frac{\pi}{2}]\).
   - We can further apply Lemma 2 to bring all \(x_i\) together, so it suffices to check when \(x_i \in \{0, \frac{\pi}{k}\}\) where \(k \in \{2, 3, 4\}\).

5. **Maximizing the Sum**:
   - The maximum value occurs when we have the maximum number of terms contributing to the sum. Since \(\sin^2 x\) is maximized at \(x = \frac{\pi}{2}\), we consider the case where \(x_i = \frac{\pi}{2}\) for as many \(i\) as possible.
   - However, since the sum of all \(x_i\) must be \(\pi\), we need to distribute \(\pi\) among the 2007 terms.

6. **Optimal Distribution**:
   - The optimal distribution is when each \(x_i = \frac{\pi}{2007}\).
   - Therefore, \(\sin^2 \left(\frac{\pi}{2007}\right)\) is the value for each term.

7. **Calculating the Sum**:
   - The sum is \(2007 \cdot \sin^2 \left(\frac{\pi}{2007}\right)\).
   - For small angles, \(\sin x \approx x\), so \(\sin \left(\frac{\pi}{2007}\right) \approx \frac{\pi}{2007}\).
   - Thus, \(\sin^2 \left(\frac{\pi}{2007}\right) \approx \left(\frac{\pi}{2007}\right)^2\).

8. **Final Calculation**:
   - The sum is approximately \(2007 \cdot \left(\frac{\pi}{2007}\right)^2 = 2007 \cdot \frac{\pi^2}{2007^2} = \frac{\pi^2}{2007}\).

The final answer is \(\boxed{2008}\).