(1) Find the possible number of roots for the equation $|x + 1| + |x + 2| + |x + 3| = a$, where $x \in R$ and $a$ is parameter.

(2) Let $\{ a_1, a_2, \ldots, a_n \}$ be an arithmetic progression, $n \in \mathbb{N}$, and satisfy the condition 
\[ \sum^{n}_{i=1}|a_i| = \sum^{n}_{i=1}|a_{i} + 1| = \sum^{n}_{i=1}|a_{i} - 2| = 507. \]
Find the maximum value of $n$.
### Part 1: Finding the possible number of roots for the equation \( |x + 1| + |x + 2| + |x + 3| = a \)

1. **Identify the critical points**:
   The function \( f(x) = |x + 1| + |x + 2| + |x + 3| \) changes its form at the points where the arguments of the absolute values are zero. These points are \( x = -1 \), \( x = -2 \), and \( x = -3 \).

2. **Analyze the function in different intervals**:
   - For \( x \leq -3 \):
     \[
     f(x) = -(x + 1) - (x + 2) - (x + 3) = -3x - 6
     \]
   - For \( -3 < x \leq -2 \):
     \[
     f(x) = -(x + 1) - (x + 2) + (x + 3) = -x
     \]
   - For \( -2 < x \leq -1 \):
     \[
     f(x) = -(x + 1) + (x + 2) + (x + 3) = x + 4
     \]
   - For \( x > -1 \):
     \[
     f(x) = (x + 1) + (x + 2) + (x + 3) = 3x + 6
     \]

3. **Determine the range of \( f(x) \)**:
   - As \( x \to -\infty \), \( f(x) \to \infty \).
   - As \( x \to \infty \), \( f(x) \to \infty \).
   - At \( x = -3 \), \( f(x) = 3 \).
   - At \( x = -2 \), \( f(x) = 2 \).
   - At \( x = -1 \), \( f(x) = 4 \).

4. **Find the possible values of \( a \)**:
   - For \( a \geq 6 \), there are two roots because the function is increasing on both sides of \( x = -1 \).
   - For \( 2 \leq a < 6 \), there is one root because the function is strictly decreasing and then strictly increasing.
   - For \( a < 2 \), there are no roots because the minimum value of \( f(x) \) is 2.

### Part 2: Finding the maximum value of \( n \)

1. **Given conditions**:
   \[
   \sum_{i=1}^{n} |a_i| = \sum_{i=1}^{n} |a_i + 1| = \sum_{i=1}^{n} |a_i - 2| = 507
   \]

2. **Properties of the function**:
   - The function \( f(x) = \sum_{i=1}^{n} |x - a_i| \) is piecewise linear and changes slope at each \( a_i \).

3. **Arithmetic progression**:
   - Let \( a_1, a_2, \ldots, a_n \) be an arithmetic progression with the first term \( a_1 \) and common difference \( d \).
   - The sequence is symmetric around its median if \( n \) is odd, or around the midpoint of the two central terms if \( n \) is even.

4. **Symmetry and equal sums**:
   - Since \( \sum_{i=1}^{n} |a_i| = \sum_{i=1}^{n} |a_i + 1| = \sum_{i=1}^{n} |a_i - 2| \), the sequence must be symmetric around 0, -1, and 2.
   - This implies that the common difference \( d \) must be at least 3 to accommodate the symmetry.

5. **Calculate the maximum \( n \)**:
   - The sum of the absolute values of the terms in the sequence must be 507.
   - For an arithmetic sequence with common difference \( d \geq 3 \), the terms are \( a_1, a_1 + d, a_1 + 2d, \ldots, a_1 + (n-1)d \).
   - The sum of the absolute values is maximized when the sequence is symmetric around 0.

6. **Determine the maximum \( n \)**:
   - If \( n = 2k \), then the sequence is symmetric around 0.
   - The sum of the absolute values is \( 3k^2 \leq 507 \).
   - Solving \( 3k^2 \leq 507 \) gives \( k \leq 13 \).
   - Therefore, \( n \leq 26 \).

The final answer is \( \boxed{ n \leq 26 } \).