In practical applications, there are some constrained multi-objective optimization problems where the constraints are either unknown or non-quantifiable, called CMOP-UC. The lack of constraint information presents significant challenges to existing evolutionary algorithms, leading to poor diversity and a high risk of premature convergence. This issue primarily arises because most of existing algorithms depend heavily on explicit constraint information to guide their search. To solve this issue, we develop a network-assisted evolutionary multitasking framework (NA-EMT). In the NA-EMT, the main task solves the original CMOP-UC by two new strategies, i.e., the perception strategy of infeasible solution value and the constraint dominance principle with predicted value. This enables the algorithm to identify valuable infeasible solutions, thereby enhancing its overall performance. The auxiliary task solves the corresponding unconstrained multi-objective optimization problem, aiming to promote convergence while preserving the diversity of solutions. Experimental results demonstrate that NA-EMT significantly outperforms 6 comparison algorithms, proving it to be an effective method for addressing CMOP-UC problems.