
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  $EFGH$ ,  $EFDC$ , and  $EHBC$  be three adjacent square faces of a cube, for which  $EC=8$ , and let  $A$  be the eighth vertex of the cube. Let  $I$ ,  $J$ , and  $K$ , be the points on  $\overline{EF}$ ,  $\overline{EH}$ , and  $\overline{EC}$ , respectively, so that  $EI=EJ=EK=2$ . A solid  $S$  is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to  $\overline{AE}$ , and containing the edges,  $\overline{IJ}$ ,  $\overline{JK}$ , and  $\overline{KI}$ . The surface area of  $S$ , including the walls of the tunnel, is  $m+n\sqrt{p}$ , where  $m$ ,  $n$ , and  $p$  are positive integers and  $p$  is not divisible by the square of any prime. Find  $m+n+p$ .

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