[ Avaa Bypassed ]

class FiniteAutomaton:
    def __init__(self, start_state, dump_state=''):
        self.start_state = start_state
        self.dump_state = dump_state
        self.table = {self.dump_state: {}}
        self.unresolved_composites = []
        self.resolved_composites = []
        self.composite_memo = {}
        self.is_updated = 1

    def __setitem__(self, xxx_todo_changeme, x):
        (state, symbol) = xxx_todo_changeme
        self.table.setdefault(state, {})[symbol] = x
        self.is_updated = 1

    def add_transition(self, state, symbol, *nexts):
        if len(nexts) == 1:
            c = nexts[0]
        else:
            for n in nexts:
                if n not in self.table:
                    self.table[n] = {}

            c = self.new_composite(nexts)
        self[state, symbol] = c
        if c not in self.table:
            self.table[c] = {}

    def get_row_items(self, state):
        try:
            x = list(self.table[state].items())
        except KeyError:
            return ()
        ris = []
        for k, v in x:
            if isinstance(v, CompositeState):
                for vi in v:
                    ris.append((k, vi))
            else:
                ris.append((k, v))
        return ris

    def make_deterministic(self):
        # Resolve all unresolved composite states so we become deterministic

        while self.unresolved_composites:
            composites = self.unresolved_composites
            self.unresolved_composites = []
            for c in composites:
                ds = {}
                l = []
                for a in c:
                    for k, v in self.get_row_items(a):
                        ds.setdefault(k, []).append(v)
                for k, v in list(ds.items()):
                    for a in c:
                        if k not in self.table[a]:
                            v.append(self.dump_state)
                            break
                for k, v in list(ds.items()):
                    self.add_transition(c, k, *v)

    def get_all_input_symbols(self):
        syms = {}
        for state, trans in list(self.table.items()):
            for k, v in trans:
                syms[k] = 1
        return syms

    def get_all_states(self):
        return self.table

    def get_all_final_states(self):
        return {}

    def get_composites(self):
        if self.is_updated:
            self.make_deterministic()
        return list(self.composite_memo.values())

    def get_transition_classes(self):
        # Get classes of states that have the same outgoing transitions
        tc = {}
        tck = {}
        for k, v in list(self.table.items()):
            trans = list(v.keys())
            trans.sort()
            trans = tuple(trans)
            ks = tc.get(trans)
            if ks is None:
                ks = []
                tc[trans] = ks
            ks.append(k)
            tck[k] = ks
        return tc, tck

    def get_minimized_dfa(self, finals):
        def markall(finals, tcv, table):
            def psrmark(l):
                for pq in l:
                    if pq in PS:
                        l1 = PS[pq]
                        del PS[pq]
                        if l1:
                            psrmark(l1)
            PS = {}
            for Q in tcv:
                for ip, p in enumerate(Q):
                    for q in Q[ip+1:]:
                        if (p in finals) == (q in finals):
                            PS[(p, q)] = ()

            for pq in list(PS.keys()):
                p, q = pq
                dp = table[p]
                dq = table[q]
                # Now we know they contain the same symbols
                dps = []
                for a in dp:
                    dpa = dp[a]
                    dqa = dq[a]
                    if dpa is not dqa:
                        dpadqa = (dpa, dqa)
                        if dpadqa in PS:
                            dps.append(dpadqa)
                        else:
                            dqadpa = (dqa, dpa)
                            if dqadpa in PS:
                                dps.append(dqadpa)
                            else:
                                l = PS[pq]
                                del PS[pq]
                                if l is not ():
                                    psrmark(l)
                                break
                else:
                    for dpadqa in dps:
                        l = PS[dpadqa]
                        if l is ():
                            l = [pq]
                            PS[dpadqa] = l
                        else:
                            l.append(pq)
            return PS

        def combine(QS, PQS):
            eqs = {}
            for Q in QS:
                for p in Q:
                    eqs[p] = [p]

            for p, q in PQS:
                #
                # Combine to equivalence classes
                #
                # Now we know that p, q are combinable
                #
                ep = eqs[p]
                eq = eqs[q]
                if eq is not ep:
                    if len(eq) > len(ep):
                        eq, ep = ep, eq
                    ep.extend(eq)
                    for q in eq:
                        eqs[q] = ep

            return eqs

        def finalize(eqs):
            csbyid = {}         # Composite state by identity of atomic state list
            csbyas = {}         # Composite state by atomic state

            i = 0
            for eq in list(eqs.values()):
                ideq = id(eq)
                if ideq not in csbyid:
                    c = 'MS%d' % i
                    i += 1
                    csbyid[ideq] = eq[0], c
                    for s in eq:
                        csbyas[s] = c

            fsa = self.__class__(csbyas[self.start_state])
            fsa.final_states = {}

            for cs0, cst in list(csbyid.values()):
                fsa.table[cst] = trans = {}
                for a, s in list(self.table[cs0].items()):
                    trans[a] = csbyas[s]
                if cs0 in finals:
                    fsa.final_states[cst] = 1

            return fsa

        if self.is_updated:
            self.make_deterministic()
            tctck = self._tctck = self.get_transition_classes()
            self.is_updated = 0
        else:
            tctck = self._tctck

        tc, tck = tctck
        tcv = list(tc.values())
        PS = markall(finals, tcv, self.table)
        eqs = combine(tcv, PS)
        fsa = finalize(eqs)
        return fsa

    def new_composite(self, args):
        cs = CompositeState(list(dict([(arg, 1) for arg in args]).keys()))
        if len(cs) == 1:
            return args[0]
        try:
            return self.composite_memo[cs]
        except KeyError:
            self.composite_memo[cs] = cs
            self.unresolved_composites.append(cs)
            return cs

    def pp(self):
        ks = list(self.table.keys())
        ks.sort()
        num = dict([(s, i) for i, s in enumerate(ks)])
        for s in ks:
            k = self.table[s]
            print('%d: %s' % (num[s], s))
            cs = list(k.keys())
            cs.sort()
            for c in cs:
                v = k[c]
                print('   %r  -> #%d: %s' % (c, num[v], v))


class CompositeState(tuple):
    pass