
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
Let the complex number $z_{1}=2+i$ correspond to the vector $\overrightarrow{AB}$ in the complex plane, the complex number $z_{2}=-1+\lambda i$ correspond to the vector $\overrightarrow{BE}$, and the complex number $z_{3}=-2+i$ correspond to the vector $\overrightarrow{EC}$, with points $A$, $E$, and $C$ lying on the same line.<br/>$(1)$ Find the value of the real number $\lambda$;<br/>$(2)$ Find the coordinates of vector $\overrightarrow{BC}$;<br/>$(3)$ Given point $D(3,5)$, if points $A$, $B$, $C$, and $D$ form a parallelogram in counterclockwise order, find the coordinates of point $A$.

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