
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
In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. $M$ is a point on side $AB$ such that $\overrightarrow{CM} = \lambda \overrightarrow{MP} (\lambda \in R)$ and $\overrightarrow{MP} = \frac{\overrightarrow{CA}}{|\overrightarrow{CA}|\cos A} + \frac{\overrightarrow{CB}}{|\overrightarrow{CB}|\cos B}$. It is also given that $|\overrightarrow{CM}| = \frac{c}{2}$ and $a^2 + b^2 = 2\sqrt{2}ab$. Find the value of angle $C$.

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