The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $ 1$ to $ 15$ in clockwise order. Committee rules state that a Martian must occupy chair $ 1$ and an Earthling must occupy chair $ 15$. Furthermore, no Earthling can sit immediately to the left of a Martian, no Martian can sit immediately to the left of a Venusian, and no Venusian can sit immediately to the left of an Earthling. The number of possible seating arrangements for the committee is $ N\cdot (5!)^3$. Find $ N$.
1. **Understanding the Problem:**
   We need to find the number of possible seating arrangements for a committee of 15 members (5 Martians, 5 Venusians, and 5 Earthlings) around a round table with specific constraints:
   - A Martian must occupy chair 1.
   - An Earthling must occupy chair 15.
   - No Earthling can sit immediately to the left of a Martian.
   - No Martian can sit immediately to the left of a Venusian.
   - No Venusian can sit immediately to the left of an Earthling.

2. **Transforming the Problem:**
   Since the table is round and the positions of the Martian in chair 1 and the Earthling in chair 15 are fixed, we can transform the problem into a linear arrangement problem with the same constraints.

3. **Defining the Constraints:**
   - The sequence must start with a Martian (M) and end with an Earthling (E).
   - The transitions between different races must follow the rules: \(MV, VE, EM\).

4. **Case Analysis:**
   We will analyze the problem by considering the number of complete cycles of \(MVE\) sequences.

   **Case 1:** \(MVE\)
   - This sequence is not possible because it only accounts for 3 members, and we need to place 15 members.

   **Case 2:** \(MVEMVE\)
   - This sequence is also not possible because it only accounts for 6 members.

   **Case 3:** \(MVEMVEMVE\)
   - This sequence is also not possible because it only accounts for 9 members.

   **Case 4:** \(MVEMVEMVEMVE\)
   - This sequence is also not possible because it only accounts for 12 members.

   **Case 5:** \(MVEMVEMVEMVEMVE\)
   - This sequence accounts for all 15 members.

5. **Counting the Arrangements:**
   For the sequence \(MVEMVEMVEMVEMVE\), we need to distribute 5 members of each race into 5 groups:
   - Each group must contain exactly one member of each race.
   - The number of ways to distribute 5 members of each race into 5 groups is given by the multinomial coefficient \(\binom{5}{1,1,1,1,1} = 5!\).

6. **Calculating the Total Number of Arrangements:**
   - Since there are 5 groups and each group can be permuted in \(5!\) ways, the total number of arrangements is \((5!)^3\).

7. **Finding \(N\):**
   - The total number of possible seating arrangements is given by \(N \cdot (5!)^3\).
   - From the case analysis, we have only one valid sequence, so \(N = 1\).

The final answer is \(\boxed{1}\).