
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
Two masses are connected with spring constant  $k$ . The masses have magnitudes  $m$  and  $M$ . The center-of-mass of the system is fixed. If  $ k = \text {100 N/m} $  and  $m=\dfrac{1}{2}M=\text{1 kg}$ , let the ground state energy of the system be  $E$ . If  $E$  can be expressed in the form  $ a \times 10^p $  eV (electron-volts), find the ordered pair  $(a,p)$ , where  $ 0 < a < 10 $ , and it is rounded to the nearest positive integer and  $p$  is an integer. For example,  $ 4.2 \times 10^7 $  should be expressed as  $(4,7)$ . 

*(Trung Phan, 10 points)*

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