Reminds me of broccoli

Matthew has a magical Fibonacci tree, which grows in the following pattern:

• During the $1^\text{st}$ day, it only has one branch (i.e. the main trunk).
• During the $2^\text{nd}$ day, it grows $F_{2+1}$ branches on the initial 1 branch.
• During the $3^\text{rd}$ day, it grows $F_{3+1}$ branches on each of the $F_{2+1}$ branches $($ newly grown during the $2^\text{nd}$ day $).$
• During the $(n+1)^\text{th}$ day, it grows $F_{(n+1)+1}$ branches on each of the $F_{n+1}$ branches $($ newly grown during the $n^\text{th}$ day $).$

His neighbor, Olivia, thinks that the tree is blocking her sunlight, so cuts $1$ branch on the $3^\text{rd}$ day as a start. Then she multiplies the number of branches she cuts by $x$ each following day. But no matter how many days she does this, the branches never seem to stop growing.

Find the maximum value of $x$ such that the above scenario is possible. If there is no solution, input your answer as $0$ .

Details and Assumptions:

• Olivia will start cutting from the newly grown branches. A branch is counted new if it is less than 1 day old.
• During the $3^\text{rd}$ day, she cuts $1$ newly grown branch, so Mark will be left with $6-1=5$ new branches. During the $4^\text{th}$ day, the remaining $5$ branches will each grow $F_{4+1} =5$ new branches, resulting in $25$ new branches. On the same day, Olivia cuts $x^{1}$ branches, so $25-x$ branches remain. And the pattern continues...
• If the branches newly grown on the $n^\text{th}$ day are all cut, the previous "new" branches will grow $F_{n+1}$ branches.

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Bonus: Explain why.