Integration with signum function

$\begin{matrix} \frac{1}{{2\pi {\sigma _1}{\sigma _2}}}\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {{\mathop{\rm sgn}} \left\{ {{\mathop{\rm sgn}} \left( {|{y_1}| - a} \right) + 1} \right\}} } {e^{ - \frac{{{y^2}_1}}{{2{\sigma ^2}_1}}}}{e^{ - \frac{{{y^2}_2}}{{2{\sigma ^2}_2}}}}d{y_1}d{y_2} \end{matrix}$

Please provide a closed form solution of this integral. $\\$ where, $\begin{matrix} \delta \left( . \right) \end{matrix}$ is the dirac delta function, $\begin{matrix} sgn(.)\end{matrix}$ 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}`	$\boxed{123}$