
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 that $A$ and $B$ are two moving points on the circle $O: x^2 + y^2 = 4$, and $\overrightarrow{OC} = \frac{5}{2} \overrightarrow{OA} - \frac{\sqrt{2}}{2} \overrightarrow{OB}$, $|\overrightarrow{AB}| = 2\sqrt{2}$. If $M$ is the midpoint of line segment $AB$, find the value of $\overrightarrow{OC} \cdot \overrightarrow{OM}$.

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