
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 the crank-slider mechanism (1-crank, 2-connecting rod). These mechanisms were first used in antiquity, in Roman sawmills. There, the rotation of a wheel driven by the force of falling water was converted into the reciprocating motion of the saw blade. The given crank rotates with a constant angular velocity $\omega=10 \ \text{rad/s}$. Find the equations of motion, the equation of the trajectory, and the speed of point $\mathrm{L}$ on the connecting rod, given that $\mathrm{OA}=\mathrm{AB}=90 \ \text{cm}$ and $\mathrm{AL}=\frac{1}{3} \mathrm{AB}$.

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