
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
Consider a crank-slider mechanism (1-crank, 2-rod). Crank mechanisms were first used in antiquity, notably in Roman sawmills. There, the rotation of a wheel driven by falling water was converted into the reciprocating motion of a saw blade. This crank rotates with a constant angular velocity $\omega = 10 \ \text{rad/sec}$. Find the equations of motion, the equation of the trajectory, and the velocity of point M on the rod, given $O A = A B = 90 \ \text{cm}$ and $A M = \frac{2}{3} A B$.

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