docs(docs): add ProblemBadges component, layout slot, and enrich pages with topics/video frontmatter, plus guide placeholders and topic‑landing map.

apps/docs/content/(01-linear-structures)/prefix-sum/238-product-of-array-except-self.mdx

@@ -2,13 +2,15 @@
 title: '238. Product of Array Except Self'
 description: Given an integer array nums, return an array answer such that answer[i] is equal to the product of all the elements of nums except nums[i]
 icon: dot
+topics:
+  - { name: "Array", slug: "array" }
+  - { name: "Prefix Sum", slug: "prefix-sum" }
+video: "bNvIQI2wAjk"
 sidebar:
   label: 'Product of Array Except Self'
   badge: 'Medium'
 ---
 
-<Badge variant="accent">Array</Badge>
-
 ::::warning
 You must write an algorithm that runs in O(n) time and without using the division operation.
 Can you solve the problem in O(1) extra space complexity? (The output array does not count as extra space for space complexity analysis.)
@@ -63,7 +65,3 @@ var productExceptSelf = function(nums) {
   return answer;
 };
 ```
-
-## Explanation
-
-<YouTube id="bNvIQI2wAjk" title="Product of Array Except Self - Leetcode 238 - Python" />

apps/docs/content/(01-linear-structures)/prefix-sum/303-range-sum-query-immutable.mdx

@@ -2,13 +2,16 @@
 title: '303. Range Sum Query - Immutable'
 description: Given an integer array nums, handle multiple queries of the following type
 icon: dot
+topics:
+  - { name: "Array", slug: "array" }
+  - { name: "Design", slug: "design" }
+  - { name: "Prefix Sum", slug: "prefix-sum" }
+video: "2pndAmo_sMA"
 sidebar:
   label: 'Range Sum Query - Immutable'
   badge: 'Easy'
 ---
 
-<Badge variant="accent">Array</Badge>
-
 ### Example 1:
 - Input: `["NumArray", "sumRange", "sumRange", "sumRange"]` `[[[-2, 0, 3, -5, 2, -1]], [0, 2], [2, 5], [0, 5]]`
 - Output: `[null, 1, -1, -3]`
@@ -66,7 +69,3 @@ class NumArray {
  * var param_1 = obj.sumRange(left,right)
  */
 ```
-
-## Explanation
-
-<YouTube id="2pndAmo_sMA" title="Range Sum Query Immutable - Leetcode 303 - Python" />

apps/docs/content/(01-linear-structures)/prefix-sum/560-subarray-sum-equals-k.mdx

@@ -2,13 +2,16 @@
 title: '560. Subarray Sum Equals K'
 description: Given an array of integers nums and an integer k, return the total number of subarrays whose sum equals to k
 icon: dot
+topics:
+  - { name: "Array", slug: "array" }
+  - { name: "Hash Table", slug: "hash-table" }
+  - { name: "Prefix Sum", slug: "prefix-sum" }
+video: "fFVZt-6sgyo"
 sidebar:
   label: 'Subarray Sum Equals K'
   badge: 'Medium'
 ---
 
-<Badge variant="accent">Array</Badge>
-
 ### Example 1:
 - Input: `nums = [1,1,1], k = 2`
 - Output: `2`
@@ -65,7 +68,3 @@ var subarraySum = function(nums, k) {
   return count
 };
 ```
-
-## Explanation
-
-<YouTube id="fFVZt-6sgyo" title="Subarray Sum Equals K - Prefix Sums - Leetcode 560 - Python" />