I think you can use recursive to find it.

func<void> find_lcs(string s, string t)
{
...
}

func<void> output(string str)
{
...
}

func<bool> find_min_lcs(int p1 = 0, int p2 = 0, int len = 0, string str)
{
if (len) = (max_lcs_len) return output(str), true;
if (p1 > s.size && p2 > t.size) return false;

For every possible element (e) sorted in order (like 'a' -> 'z')
-- For every char at [i] in string ([i] = p1 -> s.last)
---- if (e) = (s[i])
------ For every char at [j] in string ([j] = p2 -> t.last)
-------- if (e) = (t[j])
---------- if (dp_lcs[i][j] = len)
------------ if (find_min_lcs(i + 1, j + 1, len + 1, str + e))
-------------- return true;

return false;
}

then you call it

find_lcs(s, t);
find_min_lcs();