
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
Inside the  $7\times8$  rectangle below, one point is chosen a distance  $\sqrt2$  from the left side and a distance  $\sqrt7$  from the bottom side. The line segments from that point to the four vertices of the rectangle are drawn. Find the area of the shaded region.
[asy]
import graph;
size(4cm);
pair A = (0,0);
pair B = (9,0);
pair C = (9,7);
pair D = (0,7);
pair P = (1.5,3);
draw(A--B--C--D--cycle,linewidth(1.5));
filldraw(A--B--P--cycle,rgb(.76,.76,.76),linewidth(1.5));
filldraw(C--D--P--cycle,rgb(.76,.76,.76),linewidth(1.5));
[/asy]

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