Logic locking is a hardware security technique to intellectual property (IP) against security threats in the IC supply chain, especially untrusted fabs. Such techniques incorporate additional locking circuitry within an IC that induces incorrect functionality when an incorrect key is provided. The amount of error induced is known as the effectiveness of the locking technique. "SAT attacks" provide a strong mathematical formulation to find the correct key of locked circuits. In order to achieve high SAT resilience(i.e. complexity of SAT attacks), many conventional logic locking schemes fail to inject sufficient error into the circuit. For example, in the case of SARLock and Anti-SAT, there are usually very few (or only one) input minterms that cause any error at the circuit output. The state-of-the-art stripped functionality logic locking (SFLL) technique introduced a trade-off between SAT resilience and effectiveness. In this work, we prove that such a trade-off is universal in logic locking. In order to attain high effectiveness of locking without compromising SAT resilience, we propose a novel logic locking scheme, called Strong Anti-SAT (SAS). In addition to SAT attacks, removal-based attacks are also popular against logic locking. Based on SAS, we propose Robust SAS (RSAS) which is resilient to removal attacks and maintains the same SAT resilience and as effectiveness as SAS. SAS and RSAS have the following significant improvements over existing techniques. (1) SAT resilience of SAS and RSAS against SAT attack is not compromised by increase in effectiveness. (2) In contrast to prior work focusing solely on the circuit-level locking impact, we integrate SAS-locked modules into a processor and show that SAS has a high application-level impact. (3) Our experiments show that SAS and RSAS exhibit better SAT resilience than SFLL and have similar effectiveness.