
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
A four-digit number  $YEAR$  is called *very good* if the system
\begin{align*}
Yx+Ey+Az+Rw& =Y
Rx+Yy+Ez+Aw & = E\
Ax+Ry+Yz+Ew & = A
Ex+Ay+Rz+Yw &= R
\end{align*}
of linear equations in the variables  $x,y,z$  and  $w$  has at least two solutions. Find all very good  $YEAR$ s in the 21st century.

(The  $21$ st century starts in  $2001$  and ends in  $2100$ .)

*Proposed by Tomáš Bárta, Charles University, Prague*

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