
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
Given that $\overrightarrow{e_1}, \overrightarrow{e_2}$ are two non-collinear non-zero vectors in a plane, $\overrightarrow{AB}=2 \overrightarrow{e_1}+ \overrightarrow{e_2}$, $\overrightarrow{BE}=- \overrightarrow{e_1}+\lambda \overrightarrow{e_2}$, $\overrightarrow{EC}=-2 \overrightarrow{e_1}+ \overrightarrow{e_2}$, and points $A$, $E$, $C$ are collinear  
$(1)$ Find the value of the real number $\lambda$; if $\overrightarrow{e_1}=(2,1), \overrightarrow{e_2}=(2,-2)$, find the coordinates of $\overrightarrow{BC}$;  
$(2)$ Given point $D(3,6)$, under the condition of $(1)$, if points $A$, $B$, $C$, $D$ form a parallelogram $ABCD$, find the coordinates of point $A$.

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