
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 spherical shell of mass  $M$  and radius  $R$  is completely filled with a frictionless fluid, also of mass M. It is released from rest, and then it rolls without slipping down an incline that makes an angle  $\theta$  with the horizontal. What will be the acceleration of the shell down the incline just after it is released? Assume the acceleration of free fall is  $g$ . The moment of inertia of a thin shell of radius  $r$  and mass  $m$  about the center of mass is  $I = \frac{2}{3}mr^2$ ; the moment of inertia of a solid sphere of radius r and mass m about the center of mass is  $I = \frac{2}{5}mr^2$ .

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