Question on application of Chinese Remainder Theorem

$x\phantom{\rule{thickmathspace}{0ex}}\equiv \phantom{\rule{thickmathspace}{0ex}}3\phantom{\rule{thickmathspace}{0ex}}\left({\textstyle \text{mod}}\phantom{\rule{thickmathspace}{0ex}}30\right)$

$x\phantom{\rule{thickmathspace}{0ex}}\equiv \phantom{\rule{thickmathspace}{0ex}}5\phantom{\rule{thickmathspace}{0ex}}\left({\textstyle \text{mod}}\phantom{\rule{thickmathspace}{0ex}}56\right)$

I have a system of modular equation that I want to solve. However, I thought that this system has no solution because the modulos are not coprime. Further, attempting to solve using chinese remainder theorem:

$x\phantom{\rule{thickmathspace}{0ex}}\equiv \phantom{\rule{thickmathspace}{0ex}}56p\phantom{\rule{thickmathspace}{0ex}}+\phantom{\rule{thickmathspace}{0ex}}30q$

where $p$ is such that

$56p\phantom{\rule{thickmathspace}{0ex}}\equiv \phantom{\rule{thickmathspace}{0ex}}3\phantom{\rule{thickmathspace}{0ex}}\left({\textstyle \text{mod}}\phantom{\rule{thickmathspace}{0ex}}30\right)\phantom{\rule{thickmathspace}{0ex}}$

and $q$ is such that

$30q\phantom{\rule{thickmathspace}{0ex}}\equiv \phantom{\rule{thickmathspace}{0ex}}5\phantom{\rule{thickmathspace}{0ex}}\left({\textstyle \text{mod}}\phantom{\rule{thickmathspace}{0ex}}56\right)\phantom{\rule{thickmathspace}{0ex}}$

However, again the modular inverses of these do not exist.

Yet, one solution to this system of modular inequalities is 1293. How come the Chinese remainder theorem gives that no solution exists?

$x\phantom{\rule{thickmathspace}{0ex}}\equiv \phantom{\rule{thickmathspace}{0ex}}3\phantom{\rule{thickmathspace}{0ex}}\left({\textstyle \text{mod}}\phantom{\rule{thickmathspace}{0ex}}30\right)$

$x\phantom{\rule{thickmathspace}{0ex}}\equiv \phantom{\rule{thickmathspace}{0ex}}5\phantom{\rule{thickmathspace}{0ex}}\left({\textstyle \text{mod}}\phantom{\rule{thickmathspace}{0ex}}56\right)$

I have a system of modular equation that I want to solve. However, I thought that this system has no solution because the modulos are not coprime. Further, attempting to solve using chinese remainder theorem:

$x\phantom{\rule{thickmathspace}{0ex}}\equiv \phantom{\rule{thickmathspace}{0ex}}56p\phantom{\rule{thickmathspace}{0ex}}+\phantom{\rule{thickmathspace}{0ex}}30q$

where $p$ is such that

$56p\phantom{\rule{thickmathspace}{0ex}}\equiv \phantom{\rule{thickmathspace}{0ex}}3\phantom{\rule{thickmathspace}{0ex}}\left({\textstyle \text{mod}}\phantom{\rule{thickmathspace}{0ex}}30\right)\phantom{\rule{thickmathspace}{0ex}}$

and $q$ is such that

$30q\phantom{\rule{thickmathspace}{0ex}}\equiv \phantom{\rule{thickmathspace}{0ex}}5\phantom{\rule{thickmathspace}{0ex}}\left({\textstyle \text{mod}}\phantom{\rule{thickmathspace}{0ex}}56\right)\phantom{\rule{thickmathspace}{0ex}}$

However, again the modular inverses of these do not exist.

Yet, one solution to this system of modular inequalities is 1293. How come the Chinese remainder theorem gives that no solution exists?