
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
In the diagram below, square  $ABCD$  with side length 23 is cut into nine rectangles by two lines parallel to  $\overline{AB}$  and two lines parallel to  $\overline{BC}$ . The areas of four of these rectangles are indicated in the diagram. Compute the largest possible value for the area of the central rectangle.

[asy]
size(250);
 defaultpen (linewidth (0.7) + fontsize (10));
		 draw ((0,0)--(23,0)--(23,23)--(0,23)--cycle);
		 label(" $A$ ", (0,23), NW);
		 label(" $B$ ", (23, 23), NE);
		 label(" $C$ ", (23,0), SE);
		 label(" $D$ ", (0,0), SW);
		 draw((0,6)--(23,6));
		 draw((0,19)--(23,19));
		 draw((5,0)--(5,23));
		 draw((12,0)--(12,23));
		 label("13", (17/2, 21));
		 label("111",(35/2,25/2));
		 label("37",(17/2,3));
		 label("123",(2.5,12.5));[/asy]

*Proposed by Lewis Chen*

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