
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 right regular hexagonal prism has bases  $ABCDEF$ ,  $A'B'C'D'E'F'$  and edges  $AA'$ ,  $BB'$ ,  $CC'$ ,  $DD'$ ,  $EE'$ ,  $FF'$ , each of which is perpendicular to both hexagons. The height of the prism is  $5$  and the side length of the hexagons is  $6$ . The plane  $P$  passes through points  $A$ ,  $C'$ , and  $E$ . The area of the portion of  $P$  contained in the prism can be expressed as  $m\sqrt{n}$ , where  $n$  is not divisible by the square of any prime. Find  $m+n$ .

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