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Javascript, 172 bytes

Using Javascript: Try it out on any of consoles: To get the output as designed in Question..Paste into your console to run.  

for(y=n=0,s=`()()()()()()
`;y<10;y++,s+=`|
`)for(x=-2;x++<9;)s+=x>y?(Math.PI+'2384626433832795028841971693' [n++]PI+'2384626433832795028841971693'[n++]:`\\${y>8?'__':x+1|y>1?'::':'||'}\\`[y-x]||' ';console.log(s)

Using Javascript: Try it out on any of consoles: To get the output as designed in Question...  

for(y=n=0,s=`()()()()()()
`;y<10;y++,s+=`|
`)for(x=-2;x++<9;)s+=x>y?(Math.PI+'2384626433832795028841971693' [n++]:`\\${y>8?'__':x+1|y>1?'::':'||'}\\`[y-x]||' ';console.log(s)

Javascript, 172 bytes

Paste into your console to run.

for(y=n=0,s=`()()()()()()
`;y<10;y++,s+=`|
`)for(x=-2;x++<9;)s+=x>y(Math.PI+'2384626433832795028841971693'[n++]:`\\${y>8?'__':x+1|y>1?'::':'||'}\\`[y-x]||' ';console.log(s)
2 Formatted your code
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Using Javascript: Try it out on any of consoles: To get the output as designed in Question... for(y=n=0,s=()()()()()() ;y<10;y++,s+=| )for(x=-2;x++<9;)s+=x>y?(Math.PI+'2384626433832795028841971693')[n++]:\\${y>8?'__':x+1|y>1?'::':'||'}\\[y-x]||' ';console.log(s)

for(y=n=0,s=`()()()()()()
`;y<10;y++,s+=`|
`)for(x=-2;x++<9;)s+=x>y?(Math.PI+'2384626433832795028841971693' [n++]:`\\${y>8?'__':x+1|y>1?'::':'||'}\\`[y-x]||' ';console.log(s)

Using Javascript: Try it out on any of consoles: To get the output as designed in Question... for(y=n=0,s=()()()()()() ;y<10;y++,s+=| )for(x=-2;x++<9;)s+=x>y?(Math.PI+'2384626433832795028841971693')[n++]:\\${y>8?'__':x+1|y>1?'::':'||'}\\[y-x]||' ';console.log(s)

Using Javascript: Try it out on any of consoles: To get the output as designed in Question...

for(y=n=0,s=`()()()()()()
`;y<10;y++,s+=`|
`)for(x=-2;x++<9;)s+=x>y?(Math.PI+'2384626433832795028841971693' [n++]:`\\${y>8?'__':x+1|y>1?'::':'||'}\\`[y-x]||' ';console.log(s)
1
source | link

Using Javascript: Try it out on any of consoles: To get the output as designed in Question... for(y=n=0,s=()()()()()() ;y<10;y++,s+=| )for(x=-2;x++<9;)s+=x>y?(Math.PI+'2384626433832795028841971693')[n++]:\\${y>8?'__':x+1|y>1?'::':'||'}\\[y-x]||' ';console.log(s)