```ts
/**
 * Finds the cheapest combination of passes covering every trip day and reconstructs its purchases in increasing start-day order.
 */
export function cheapestPassPlan(tripDays: number[], passes: { span: number; cost: number }[]): { total: number; purchases: number[][] } {
  if (tripDays.length === 0) return { total: 0, purchases: [] };

  const n = tripDays.length;
  const dp: number[] = Array(n + 1).fill(Infinity);
  dp[0] = 0;

  for (let i = 1; i <= n; i++) {
    for (const pass of passes) {
      if (tripDays[i - 1] - tripDays[Math.max(0, i - pass.span)] + 1 >= pass.span) {
        dp[i] = Math.min(dp[i], dp[Math.max(0, i - pass.span)] + pass.cost);
      }
    }
  }

  const purchases: number[][] = [];
  let i = n;
  while (i > 0) {
    for (const pass of passes) {
      if (tripDays[i - 1] - tripDays[Math.max(0, i - pass.span)] + 1 >= pass.span && dp[i] === dp[Math.max(0, i - pass.span)] + pass.cost) {
        purchases.unshift([tripDays[Math.max(0, i - pass.span)], pass.span]);
        i -= pass.span;
        break;
      }
    }
  }

  return { total: dp[n], purchases };
}
```