<p>You are given an array <code>prices</code> where <code>prices[i]</code> is the price of a given stock on the <code>i<sup>th</sup></code> day.</p>

<p>Find the maximum profit you can achieve. You may complete <strong>at most two transactions</strong>.</p>

<p><strong>Note:</strong> You may not engage in multiple transactions simultaneously (i.e., you must sell the stock before you buy again).</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> prices = [3,3,5,0,0,3,1,4]
<strong>Output:</strong> 6
<strong>Explanation:</strong> Buy on day 4 (price = 0) and sell on day 6 (price = 3), profit = 3-0 = 3.
Then buy on day 7 (price = 1) and sell on day 8 (price = 4), profit = 4-1 = 3.</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> prices = [1,2,3,4,5]
<strong>Output:</strong> 4
<strong>Explanation:</strong> Buy on day 1 (price = 1) and sell on day 5 (price = 5), profit = 5-1 = 4.
Note that you cannot buy on day 1, buy on day 2 and sell them later, as you are engaging multiple transactions at the same time. You must sell before buying again.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> prices = [7,6,4,3,1]
<strong>Output:</strong> 0
<strong>Explanation:</strong> In this case, no transaction is done, i.e. max profit = 0.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= prices.length &lt;= 10<sup>5</sup></code></li>
	<li><code>0 &lt;= prices[i] &lt;= 10<sup>5</sup></code></li>
</ul>



class Solution:
    def maxProfit(self, prices: List[int]) -> int:
        levels = [{
            "I": {}
        }]
        
        for i in range(len(prices) + 1):
            levels.append({})
            curr_level = levels[i]
            if i == len(prices):
                levels[-1] = {"E":{}}
            for key in curr_level:
                if i == len(prices):
                    if "E" not in levels[i + 1]:
                        levels[i + 1]["E"] = {key: 0}
                    else:
                        levels[i + 1]["E"][key] = 0
                    continue
                if key in ("I", "v0"):
                    if "b1" not in levels[i + 1]:
                        levels[i + 1]["b1"] = {key: -prices[i]}
                    else:
                        levels[i + 1]["b1"][key] = -prices[i]
                    if "v0" not in levels[i + 1]:
                        levels[i + 1]["v0"] = {key: 0}
                    else:
                        levels[i + 1]["v0"][key] = 0
                if key in ("b1", "k1"):
                    if "s1" not in levels[i + 1]:
                        levels[i + 1]["s1"] = {key: prices[i]}
                    else:
                        levels[i + 1]["s1"][key] = prices[i]
                    if "k1" not in levels[i + 1]:
                        levels[i + 1]["k1"] = {key: 0} 
                    else:
                        levels[i + 1]["k1"][key] = 0
                if key in ("s1", "v1"):
                    if "v1" not in levels[i + 1]:
                        levels[i + 1]["v1"] = {key: 0} 
                    else:
                        levels[i + 1]["v1"][key] = 0
                    if "b2" not in levels[i + 1]:
                        levels[i + 1]["b2"] = {key: -prices[i]} 
                    else:
                        levels[i + 1]["b2"][key] = -prices[i]
                if key in ("b2", "k2"):
                    if "s2" not in levels[i + 1]:
                        levels[i + 1]["s2"] = {key: prices[i]} 
                    else:
                        levels[i + 1]["s2"][key] = prices[i]
                    if "k2" not in levels[i + 1]:
                        levels[i + 1]["k2"] = {key: 0} 
                    else:
                        levels[i + 1]["k2"][key] = 0
                if key in ("s2", "v2"):
                    if "v2" not in levels[i + 1]:
                        levels[i + 1]["v2"] = {key: 0} 
                    else:
                        levels[i + 1]["v2"][key] = 0
        for i in range(len(levels)):
            if i == 0:
                levels[0]["I"]["val"] = 0
                continue
            for key in levels[i]:
                max_ = -float(inf)
                for src in levels[i][key]:
                    tot = levels[i][key][src] + levels[i - 1][src]["val"]
                    if tot > max_:
                        max_ = tot
                levels[i][key]["val"] = max_
        return levels[-1]["E"]["val"]
                    
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