
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
Given two spheres, the larger one has a mass of $M$, and the smaller one has a mass of $m$. Let point $A$ be inside the line segment connecting the centers of the two spheres, and point $B$ be outside this segment, chosen such that both spheres exert an equal gravitational force on a unit mass placed at $B$ according to Newton's law of gravitation. If the distance between points $A$ and $B$ is $d$, determine the distance between the centers of the spheres. What is this distance if $M=2116 \mathrm{~kg}$, $m=16 \mathrm{~kg}$, and $d=1.15 \mathrm{~m}$?

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