Question 100
The JMO Cluster initially consists of five stars $O$, $A$, $B$, $C$, and $D$. Each star is assigned a value called its **importance**. The importance of $O$ is $0$, and the importance of each of $A$, $B$, $C$, and $D$ is $1$. Moreover, there are one-way direct flights from $O$ to $A$ and $C$; from $A$ to $B$ and $D$; from $B$ to $O$; from $C$ to $B$ and $D$; and from $D$ to $O$, and no other direct flights exist.
To prevent the stars from aging, the JMO Cluster periodically performs an operation consisting of the following sequence of actions:
(1) Abolish all existing direct flights and destroy all the stars.
(2) For each direct flight $f$ abolished in step (1), construct a new star $S_f$. Then assign to $S_f$ an importance equal to the sum of the importances of the departure and arrival stars of $f$.
(3) For every ordered pair $(f, f')$ of two abolished direct flights such that the arrival star of $f$ coincides with the departure star of $f'$, open a one-way direct flight from $S_f$ to $S_{f'}$.
Find the sum of the importances of the stars constructed in the 100th operation.
To prevent the stars from aging, the JMO Cluster periodically performs an operation consisting of the following sequence of actions:
(1) Abolish all existing direct flights and destroy all the stars.
(2) For each direct flight $f$ abolished in step (1), construct a new star $S_f$. Then assign to $S_f$ an importance equal to the sum of the importances of the departure and arrival stars of $f$.
(3) For every ordered pair $(f, f')$ of two abolished direct flights such that the arrival star of $f$ coincides with the departure star of $f'$, open a one-way direct flight from $S_f$ to $S_{f'}$.
Find the sum of the importances of the stars constructed in the 100th operation.