The correct word to be placed in the blanks would be 'cautiously'.
The way Allisa prepares for the match shows her extreme desire to win the match. She even researches about her opponent and studies his way of playing the game too. She has not left any stones unturned in her struggle to win the game. Her research about her opponent Lex is a secret about which Lex would never know. These circumstances reveal the cautious nature of Allisa for winning the match.