From a certain point on your island, you notice a somewhat nearby island at a distance of $10$ km$.$ Perhaps it is inhabited, so you would like to transmit an alert message and see if there is a response.

After finding more parts in your backpack, you construct a basic wireless communication system which can also transmit data. Despite the challenges of an unshielded environment prone to interference, you aim for a data transmission rate of up to $1$ Mbps$.$ Through some resourceful experimentation, you discover that your wireless setup supports frequencies of up to $0\mathrm{.}5$ MHz$.$ Exceeding this could lead to intersymbol interference, corrupting your data.

To tackle this, you decide to use a multilevel bipolar NRZ modulation scheme, achieving the required bitrate within the frequency limitations of your wireless system. Additionally, your memory serves you well from your days of mastering communication systems, specifically error correction and detection with Hamming codes. To enhance the reliability of your makeshift communication system, you implement a Hamming$(12,8)$ configuration.

Upon completing your setup, you decide to test it by transmitting a simple message in binary ASCII format: "Help!" However, as fate would have it$,$ a lightning storm passes over the island, introducing noise into your wireless signal. This noise results in the flipping of $3$ bits in the transmitted message during PCM decoding, marked in red: $010011001000011001010100011011000000011100011000001000011001.$

Which of the following statements is true regarding the error$-$correcting stage?