






"""simple docstring"""






UpperCAmelCase_   : Dict   			= range(2, 20 + 1)
UpperCAmelCase_   : Union[str, Any]   			= [10**k for k in range(ks[-1] + 1)]
UpperCAmelCase_   : dict[int, dict[int, list[list[int]]]]   			= {}



def 					_A			(__a    ,				__a    ,				__a    ,				__a							) ->       Any:
    """simple docstring"""






    SCREAMING_SNAKE_CASE_      :  Dict    						=		sum(a_i[j] for j in range(__a    ,				len(__a							)							)							)
    SCREAMING_SNAKE_CASE_      :  Optional[int]    						=		sum(a_i[j] * base[j] for j in range(min(len(__a							)    ,				__a							)							)							)

    SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_      :  Union[str, Any]    						=		0, 0
    SCREAMING_SNAKE_CASE_      :  str    						=		n - i

    SCREAMING_SNAKE_CASE_      :  Dict    						=		memo.get(__a							)

    if sub_memo is not None:
        SCREAMING_SNAKE_CASE_      :  str    						=		sub_memo.get(__a							)

        if jumps is not None and len(__a							) > 0:
            # find and make the largest jump without going over
            SCREAMING_SNAKE_CASE_      :  List[str]    						=		-1
            for _k in range(len(__a							) - 1    ,				-1    ,				-1							):
                if jumps[_k][2] <= k and jumps[_k][1] <= max_dn:
                    SCREAMING_SNAKE_CASE_      :  Optional[Any]    						=		_k
                    break

            if max_jump >= 0:
                SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_      :  Union[str, Any]    						=		jumps[max_jump]
                # since the difference between jumps is cached, add c
                SCREAMING_SNAKE_CASE_      :  Optional[Any]    						=		diff + c
                for j in range(min(__a    ,				len(__a							)							)							):
                    SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_      :  Tuple    						=		divmod(__a    ,				10							)
                if new_c > 0:
                    add(__a    ,				__a    ,				__a							)

        else:
            SCREAMING_SNAKE_CASE_      :  Optional[Any]    						=		[]
    else:
        SCREAMING_SNAKE_CASE_      :  List[Any]    						=		{c: []}
        SCREAMING_SNAKE_CASE_      :  Optional[int]    						=		sub_memo

    if dn >= max_dn or c + diff >= base[k]:
        return diff, dn

    if k > ks[0]:
        while True:
            # keep doing smaller jumps
            SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_      :  Optional[Any]    						=		next_term(__a    ,				k - 1    ,				i + dn    ,				__a							)
            diff += _diff
            dn += terms_jumped

            if dn >= max_dn or c + diff >= base[k]:
                break
    else:
        # would be too small a jump, just compute sequential terms instead
        SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_      :  Union[str, Any]    						=		compute(__a    ,				__a    ,				i + dn    ,				__a							)
        diff += _diff
        dn += terms_jumped

    SCREAMING_SNAKE_CASE_      :  List[str]    						=		sub_memo[c]

    # keep jumps sorted by # of terms skipped
    SCREAMING_SNAKE_CASE_      :  List[Any]    						=		0
    while j < len(__a							):
        if jumps[j][1] > dn:
            break
        j += 1

    # cache the jump for this value digitsum(b) and c
    sub_memo[c].insert(__a    ,				(diff, dn, k)							)
    return (diff, dn)



def 					_A			(__a    ,				__a    ,				__a    ,				__a							) ->       Optional[int]:
    """simple docstring"""






    if i >= n:
        return 0, i
    if k > len(__a							):
        a_i.extend([0 for _ in range(k - len(__a							)							)]							)

    # note: a_i -> b * 10^k + c
    # ds_b -> digitsum(b)
    # ds_c -> digitsum(c)
    SCREAMING_SNAKE_CASE_      :  Any    						=		i
    SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_      :  str    						=		0, 0, 0
    for j in range(len(__a							)							):
        if j >= k:
            ds_b += a_i[j]
        else:
            ds_c += a_i[j]

    while i < n:
        i += 1
        SCREAMING_SNAKE_CASE_      :  Dict    						=		ds_c + ds_b
        diff += addend
        SCREAMING_SNAKE_CASE_      :  Tuple    						=		0
        for j in range(__a							):
            SCREAMING_SNAKE_CASE_      :  Union[str, Any]    						=		a_i[j] + addend
            SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_      :  Optional[int]    						=		divmod(__a    ,				10							)

            ds_c += a_i[j]

        if addend > 0:
            break

    if addend > 0:
        add(__a    ,				__a    ,				__a							)
    return diff, i - start_i



def 					_A			(__a    ,				__a    ,				__a							) ->       Optional[Any]:
    """simple docstring"""






    for j in range(__a    ,				len(__a							)							):
        SCREAMING_SNAKE_CASE_      :  Union[str, Any]    						=		digits[j] + addend
        if s >= 10:
            SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_      :  Optional[Any]    						=		divmod(__a    ,				10							)
            SCREAMING_SNAKE_CASE_      :  Optional[int]    						=		addend // 10 + quotient
        else:
            SCREAMING_SNAKE_CASE_      :  Tuple    						=		s
            SCREAMING_SNAKE_CASE_      :  Tuple    						=		addend // 10

        if addend == 0:
            break

    while addend > 0:
        SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_      :  str    						=		divmod(__a    ,				10							)
        digits.append(__a							)



def 					_A			(__a = 10**15							) ->       int:
    """simple docstring"""






    SCREAMING_SNAKE_CASE_      :  Optional[int]    						=		[1]
    SCREAMING_SNAKE_CASE_      :  Dict    						=		1
    SCREAMING_SNAKE_CASE_      :  Dict    						=		0
    while True:
        SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_      :  List[Any]    						=		next_term(__a    ,				20    ,				i + dn    ,				__a							)
        dn += terms_jumped
        if dn == n - i:
            break

    SCREAMING_SNAKE_CASE_      :  List[str]    						=		0
    for j in range(len(__a							)							):
        a_n += digits[j] * 10**j
    return a_n


if __name__ == "__main__":
  print(f'''{solution() = }''')










"""simple docstring"""






def 					_A			(__a = 50							) ->       int:
    """simple docstring"""






    SCREAMING_SNAKE_CASE_      :  Tuple    						=		[[0] * 3 for _ in range(length + 1							)]

    for row_length in range(length + 1							):
        for tile_length in range(2    ,				5							):
            for tile_start in range(row_length - tile_length + 1							):
                different_colour_ways_number[row_length][tile_length - 2] += (
                    different_colour_ways_number[row_length - tile_start - tile_length][
                        tile_length - 2
                    ]
                    + 1
                )

    return sum(different_colour_ways_number[length]							)


if __name__ == "__main__":
  print(f'''{solution() = }''')


