To solve this problem, we will write a Python code that computes the minimal path sum $A(N)$ for $N = 10^7$ efficiently using dynamic programming with memoization. Below is the code:def compute_sn(s1, n, mod=998388889): """ Compute the sequence s_n with s_1 given and s_n = s_{n-1}^2 % mod. """ s = [0] * (n + 1) s[1] = s1 for i in range(2, n + 1): s[i] = (s[i - 1] ** 2) % mod return s

def generate_ai_bi(s, n): """ Generate a_n = s_{2n-1} and b_n = s_{2n}. """ a = [0] * (n + 1) b = [0] * (n + 1) for i in range(1, n + 1): a[i] = s[2 * i - 1] b[i] = s[2 * i] return a, b

def minimal_path_sum(a, b, n): """ Calculate the minimal path sum A(N) for M_{i,j} = a_i + b_j. Use dynamic programming to calculate the path sum. """ # Initialize the DP table dp = [[float('inf')] * (n + 1) for _ in range(n + 1)] dp[1][1] = a[1] + b[1] # Start point

# Fill the DP table
for i in range(1, n + 1):
    for j in range(1, n + 1):
        if i > 1:
            dp[i][j] = min(dp[i][j], dp[i - 1][j] + a[i] + b[j])
        if j > 1:
            dp[i][j] = min(dp[i][j], dp[i][j - 1] + a[i] + b[j])

return dp[n][n]

Define constants and parameters

s1 = 102022661 N = 10 ** 7 mod = 998388889

Generate the s_n sequence for the necessary length

length_required = 2 * N # We need s_{2N} to calculate a_N and b_N s = compute_sn(s1, length_required, mod)

Generate a_n and b_n

a, b = generate_ai_bi(s, N)

Compute the minimal path sum A(N)

Note: For large N, this computation may require optimization.

result = minimal_path_sum(a, b, N) resultThe computation was interrupted. Since calculating this directly for $N = 10^7$ is computationally intensive and requires significant resources, we can explore an optimized approach. I will explain an alternative optimization strategy below.

Strategy:

1. Iterative Calculation: Instead of precomputing the entire sequence, generate $a_i$ and $b_i$ on the fly as needed for the dynamic programming table.
2. Space Optimization: Instead of maintaining the entire DP table, use two arrays to store the current and previous row, reducing memory usage to $O(N)$.

Would you like me to revise and implement this more optimized approach?