
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  $A_1B_1C_1D_1A_2B_2C_2D_2$  be a unit cube, with  $A_1B_1C_1D_1$  and  $A_2B_2C_2D_2$  opposite square faces, and let  $M$  be the center of face  $A_2 B_2 C_2 D_2$ . Rectangular pyramid  $MA_1B_1C_1D_1$  is cut out of the cube. If the surface area of the remaining solid can be expressed in the form  $a + \sqrt{b}$ , where  $a$  and  $b$  are positive integers and  $b$  is not divisible by the square of any prime, find  $a+b$ .

*Author: Alex Zhu*

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