Improvement in lower bound of American call with discrete dividends

Question

Suppose a stock pays 2 discrete dividends $$d_1, d_2$$ at times $$t_1, t_2$$ respectively, where $$t < t_1 < t_2 < T.$$ Assume the risk-free rate, $$r$$, is a positive constant. Given that

1. The lower and upper bounds for an American call, $$C_t$$, is trivially determined as $$\max{ \left( S_t - D_t - Ke^{-r(T-t)}, S_t - K, 0 \right)} \leq C_t \leq S_t.$$

2. Consider a strategy that exercises the option at the instant before time $$t_1$$, we have $$S_t - Ke^{-r(t_1 -t)} \leq C_t.$$

3. Consider a strategy that exercises the option at the instant before time $$t_2$$, we have $$S_t - d_1e^{-r(t_1 -t)} - Ke^{-r(t_2 -t)} \leq C_t.$$

Combining the three inequalities above, what is the improved lower bound for an American call in this situation?

My attempt

I understand that American call with discrete dividends should at most be exercised at the instant before the ex-dividend dates for optimal payoff, so I obtained the lower bound as

$$\max{\left(S_t - Ke^{-r (t_1 - t)}, S_t - d_1e^{-r (t_1 - t)} - Ke^{-r (t_2 - t)}, S_t - D_t- Ke^{-r (t_1 - t)}, 0\right)} \leq C_t$$ where $$D_t = d_1e^{-r (t_1 - t)} + d_2e^{-r (t_2 - t)}.$$

However, I am not sure if I should include $$S_t - K$$ into the max function which was given in the original question.

Any intuitive explanation will be highly appreciated!