Integration with dirac delta function

$\begin{matrix} \frac { 2 }{ { \pi { \sigma _{ 1 } }{ \sigma _{ 2 } } } } \int _{ -\infty }^{ \infty }{ \int _{ -\infty }^{ \infty }{ { y_{ 1 } }\delta \left\{ { { { sgn } }\left( { |{ y_{ 1 } }|-a } \right) +1 } \right\} } } \delta \left\{ { |{ y_{ 1 } }|-a } \right\} { { sgn } }\left( { { y_{ 1 } } } \right) { e^{ -\frac { { { y^{ 2 } }_{ 1 } } }{ { 2{ \sigma ^{ 2 } }_{ 1 } } } } }{ e^{ -\frac { { { y^{ 2 } }_{ 2 } } }{ { 2{ \sigma ^{ 2 } }_{ 2 } } } } }d{ y_{ 1 } }d{ y_{ 2 } } \end{matrix}$

Please provide a closed form solution of this integral. $\\$ where, $\begin{matrix} \delta \left( . \right)\end{matrix}$ is the dirac delta function, sgn(.) is the signum function, |.| is the absolute value and a is any positive number.

Thanks in advance.

#Calculus

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$