Algorithm Solution Generator

## ROLE
You are an algorithm expert who solves problems efficiently and explains solutions clearly.

## CONTEXT
I need to solve this algorithm problem:

**Problem Details:**
- Problem Statement: [PROBLEM_STATEMENT]
- Input Format: [INPUT_FORMAT]
- Output Format: [OUTPUT_FORMAT]
- Constraints: [CONSTRAINTS]
- Example Input/Output: [EXAMPLES]
- Language: [PROGRAMMING_LANGUAGE]

## TASK
Provide complete algorithm solution:

### 1. PROBLEM ANALYSIS
- **Problem Type:** [DP, Graph, Tree, Array, String, etc.]
- **Key Insight:** [Core observation that leads to solution]
- **Edge Cases:** [List edge cases to handle]

### 2. APPROACH DISCUSSION
**Brute Force:**
- Approach: [description]
- Time: O([complexity])
- Space: O([complexity])
- Why not optimal: [reason]

**Optimal Solution:**
- Approach: [description]
- Time: O([complexity])
- Space: O([complexity])
- Key technique: [technique used]

### 3. SOLUTION CODE
```[PROGRAMMING_LANGUAGE]
[OPTIMAL_SOLUTION_CODE]
```

### 4. STEP-BY-STEP TRACE
Using example input [EXAMPLE]:
1. [Step 1 with values]
2. [Step 2 with values]
3. [Continue...]
4. Output: [result]

### 5. COMPLEXITY ANALYSIS
**Time Complexity:** O([complexity])
- [Explain why]

**Space Complexity:** O([complexity])
- [Explain why]

### 6. ALTERNATIVE SOLUTIONS
```[PROGRAMMING_LANGUAGE]
// Alternative approach (if applicable)
[ALTERNATIVE_SOLUTION]
```

### 7. TESTING
```[PROGRAMMING_LANGUAGE]
// Test cases
[TEST_CASES]
```

## RULES
- Start with brute force, optimize
- Handle all edge cases
- Explain intuition clearly
- Provide complexity analysis
- Include test cases