Buggy python submission (verdict: Runtime Error):
# undo が出来るゲームは全て2手だけ見ればよい
ops = ["+", "*", "-", "&", "^", "|"]
numbers = ["1", "2", "3", "4", "5", "6", "7", "8", "9", "0"]


def check(x):
    # +*
    op = 0
    for c in x:
        if c in ops:
            op += 1
            if op >= 2:
                return False
        else:
            op = 0
    # +...
    for op in ops:
        if x.startswith(op):
            return False
        if ("(" + op) in x:
            return False
    # 0がないか
    zero_ok = False
    for c in x:
        if not zero_ok and c == "0":
            return False
        if c in ops:
            zero_ok = False
        elif c in numbers:
            zero_ok = True
        else:  # ( )
            zero_ok = False
    try:
        val = eval(x)
        return val
    except:
        return False


def get_nexts(x):
    # 削除
    result = []
    for i in range(len(x)):
        y = x[:i] + x[i + 1:]
        val = check(y)
        if val != False:
            result.append((val, y))
    # 追加
    for i in range(len(x) + 1):
        add_list = numbers + ops
        for s in add_list:
            y = x[:i] + s + x[i:]
            val = check(y)
            if val != False:
                result.append((val, y))
    return result


while True:
    n, x = input().split(" ")
    n = int(n)
    if n == 0:
        quit()
    nexts = get_nexts(x)
    if n == 1:
        nexts.sort(key=lambda a: - a[0])
        print(nexts[0][0])
        continue
    maxval = eval(x)
    tele = x
    minvals = []
    for (val, y) in nexts:
        nextss = get_nexts(y)
        nextss.sort(key=lambda a: a[0])
        minvals.append(nextss[0][0])
        if maxval < nextss[0][0]:
            maxval = nextss[0][0]
            tele = nextss[0][1]
    if n % 2 == 0:
        print(max(minvals))
        continue
    nexts = get_nexts(tele)
    if n % 2 == 1:
        nexts.sort(key=lambda a: -a[0])
        print(nexts[0][0])
        continue



Fixed submission:
# undo が出来るゲームは全て2手だけ見ればよい
ops = ["+", "*", "-", "&", "^", "|"]
numbers = ["1", "2", "3", "4", "5", "6", "7", "8", "9", "0"]


def check(x):
    # +*
    op = 0
    for c in x:
        if c in ops:
            op += 1
            if op >= 2:
                return None
        else:
            op = 0
    # +...
    for op in ops:
        if x.startswith(op):
            return None
        if ("(" + op) in x:
            return None
    # 0がないか
    zero_ok = False
    for c in x:
        if not zero_ok and c == "0":
            return None
        if c in numbers:
            zero_ok = True
        else:
            zero_ok = False
    try:
        return int(eval(x))
    except:
        return None


def get_nexts(x):
    # 削除
    result = []
    for i in range(len(x)):
        y = x[:i] + x[i + 1:]
        val = check(y)
        if val != None:
            result.append((val, y))
    # 追加
    for i in range(len(x) + 1):
        add_list = numbers + ops
        for s in add_list:
            y = x[:i] + s + x[i:]
            val = check(y)
            if val != None:
                result.append((val, y))
    return result


def get_nexts_val(x):
    # 削除
    result = []
    for i in range(len(x)):
        y = x[:i] + x[i + 1:]
        val = check(y)
        if val != None:
            result.append(val)
    # 追加
    for i in range(len(x) + 1):
        add_list = numbers + ops
        for s in add_list:
            y = x[:i] + s + x[i:]
            val = check(y)
            if val != None:
                result.append(val)
    return result


while True:
    n, x = input().split(" ")
    n = int(n)
    if n == 0:
        quit()
    nexts = get_nexts(x)
    if n % 2 == 1:
        nexts.sort(key=lambda a: - a[0])
        print(nexts[0][0])
        continue
    minvals = []
    for (val, y) in nexts:
        nextss = get_nexts_val(y)
        nextss.sort()
        minvals.append(nextss[0])
    print(max(minvals))