docs(content): add Merge Intervals page, quicksort guide, and bump guide badges

apps/docs/content/(05-decision-space)/intervals/56-merge-intervals.mdx

@@ -0,0 +1,50 @@
+---
+title: '56. Merge Intervals'
+description: Given an array of intervals where intervals[i] = [starti, endi], merge all overlapping intervals, and return an array of the non-overlapping intervals that cover all the intervals in the input
+icon: dot
+topics:
+  - { name: "Array", slug: "array" }
+  - { name: "Sorting", slug: "sorting" }
+  - { name: "Quicksort", slug: "quicksort" }
+sidebar:
+  label: 'Merge Intervals'
+  badge: 'Medium'
+---
+
+### Example 1:
+- Input: `intervals = [[1,3],[2,6],[8,10],[15,18]]`
+- Output: `[[1,6],[8,10],[15,18]]`
+- Explanation: Since `intervals [1,3]` and `[2,6]` overlap, merge them into `[1,6]`.
+
+### Example 2:
+- Input: `intervals = [[1,4],[4,5]]`
+- Output: `[[1,5]]`
+- Explanation: Intervals `[1,4]` and `[4,5]` are considered overlapping.
+
+### Example 3:
+- Input: `intervals = [[4,7],[1,4]]`
+- Output: `[[1,7]]`
+- Explanation: Intervals `[1,4]` and `[4,7]` are considered overlapping.
+
+### Constraints:
+
+- `1 <= intervals.length <= 10^4`
+- `intervals[i].length == 2`
+- `0 <= starti <= endi <= 10^4`
+
+## Solution
+
+```py
+class Solution:
+    def merge(self, intervals: list[list[int]]) -> list[list[int]]:
+        intervals.sort(key=lambda x: x[0])
+        result = []
+
+        for start, end in intervals:
+            if result and start <= result[-1][1]:
+                result[-1][1] = max(result[-1][1], end)
+            else:
+                result.append([start, end])
+
+        return result
+```