$300$ couples (one man, one woman) are invited to a party. Everyone at the party either always tells the truth or always lies. Exactly $2/3$ of the men say their partner always tells the truth and the remaining $1/3$ say their partner always lies. Exactly $2/3$ of the women say their partner is the same type as themselves and the remaining $1/3$ say their partner is different. Find $a$, the maximum possible number of people who tell the truth, and $b$, the minimum possible number of people who tell the truth. Express your answer as $(a,b)$.
1. **Define Variables and Constraints:**
   - Let \( M \) be the total number of men, and \( W \) be the total number of women. Here, \( M = W = 300 \).
   - Let \( T_m \) be the number of men who tell the truth, and \( L_m \) be the number of men who lie. Similarly, let \( T_w \) be the number of women who tell the truth, and \( L_w \) be the number of women who lie.
   - We know that \( T_m + L_m = 300 \) and \( T_w + L_w = 300 \).

2. **Analyze Men's Statements:**
   - \( \frac{2}{3} \) of the men say their partner always tells the truth. This implies \( \frac{2}{3} \times 300 = 200 \) men say their partner tells the truth.
   - \( \frac{1}{3} \) of the men say their partner always lies. This implies \( \frac{1}{3} \times 300 = 100 \) men say their partner lies.

3. **Analyze Women's Statements:**
   - \( \frac{2}{3} \) of the women say their partner is the same type as themselves. This implies \( \frac{2}{3} \times 300 = 200 \) women say their partner is the same type.
   - \( \frac{1}{3} \) of the women say their partner is different. This implies \( \frac{1}{3} \times 300 = 100 \) women say their partner is different.

4. **Determine Truth-Tellers and Liars:**
   - Let \( x \) be the number of truth-telling men whose partners are also truth-tellers.
   - Let \( y \) be the number of truth-telling men whose partners are liars.
   - Let \( z \) be the number of lying men whose partners are truth-tellers.
   - Let \( w \) be the number of lying men whose partners are also liars.

   We have the following equations based on the constraints:
   \[
   x + y + z + w = 300 \quad \text{(total number of couples)}
   \]
   \[
   x + y = T_m \quad \text{(total number of truth-telling men)}
   \]
   \[
   z + w = L_m \quad \text{(total number of lying men)}
   \]
   \[
   x + z = T_w \quad \text{(total number of truth-telling women)}
   \]
   \[
   y + w = L_w \quad \text{(total number of lying women)}
   \]

5. **Maximize the Number of Truth-Tellers:**
   - To maximize the number of truth-tellers, we want to maximize \( T_m \) and \( T_w \).
   - If \( T_m = 200 \) and \( T_w = 200 \), then \( L_m = 100 \) and \( L_w = 100 \).
   - This configuration satisfies the constraints, and the total number of truth-tellers is \( T_m + T_w = 200 + 200 = 400 \).

6. **Minimize the Number of Truth-Tellers:**
   - To minimize the number of truth-tellers, we want to minimize \( T_m \) and \( T_w \).
   - If \( T_m = 100 \) and \( T_w = 100 \), then \( L_m = 200 \) and \( L_w = 200 \).
   - This configuration satisfies the constraints, and the total number of truth-tellers is \( T_m + T_w = 100 + 100 = 200 \).

Therefore, the maximum number of truth-tellers is \( a = 400 \), and the minimum number of truth-tellers is \( b = 200 \). This means our answer is \(\boxed{(400,200)}\).