
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
For the $k$-th bit element, binary code is commonly used in communication. However, sometimes bit errors occur during the communication process (i.e., a bit changes from $0$ to $1$, or from $1$ to $0$).
It is known that a certain binary code $x_{1}x_{2}\ldots x_{7}$ satisfies the following set of parity check equations: $\begin{cases} x_{4}\oplus x_{5}\oplus x_{6}\oplus x_{7}=0 \\ x_{2}\oplus x_{3}\oplus x_{6}\oplus x_{7}=0 \\ x_{1}\oplus x_{3}\oplus x_{5}\oplus x_{7}=0 \end{cases}$, where the operation $\oplus$ is defined as: $0\oplus0=0$, $0\oplus1=1$, $1\oplus0=1$, $1\oplus1=0$.
Now, it is known that a binary code of this type changed to $1101101$ after a bit error occurred only at the $k$-th position during the communication process. Using the above set of parity check equations, determine the value of $k$.

Present the answer in LaTex format: \boxed{Your answer}
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