When Masha came to math classes today, she saw two integer sequences of length $\mathrm{n-1n-1 on\; the\; blackboard.\; Let\text{'}s\; denote\; the\; elements\; of\; the\; first\; sequence\; as aiai (0\le ai\le 30\le ai\le 3),\; and\; the\; elements\; of\; the\; second\; sequence\; as bibi (0\le bi\le 30\le bi\le 3).}$

Masha became interested if or not there is an integer sequence of length $\mathrm{nn,\; which\; elements\; we\; will\; denote\; as titi (0\le ti\le 30\le ti\le 3),\; so\; that\; for\; every ii (1\le i\le n-11\le i\le n-1)\; the\; following\; is\; true:}$

• ${\mathrm{ai=ti|ti+1ai=ti|ti+1 (where || denotes\; the bitwise\; OR\; operation)\; and}}^{}$
• ${\mathrm{bi=ti\&ti+1bi=ti\&ti+1 (where \&\& denotes\; the bitwise\; AND\; operation).}}^{}$

The question appeared to be too difficult for Masha, so now she asked you to check whether such a sequence ${\mathrm{titi of\; length nn exists.\; If\; it\; exists,\; find\; such\; a\; sequence.\; If\; there\; are\; multiple\; such\; sequences,\; find\; any\; of\; them.}}^{}$