
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 vectors: $\overrightarrow{a} = (\cos x, \sin x)$, $\overrightarrow{b} = (\cos y, \sin y)$, $\overrightarrow{c} = (\sin x, \cos x)$, and $|\overrightarrow{a} - \overrightarrow{b}| = \frac{2\sqrt{5}}{5}$.

(1) Find the value of $\cos(x-y)$;  
(2) If the graph of the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{c}$ is translated to the left by $m$ ($m > 0$) units to obtain the graph of function $g(x)$, which is symmetric about the y-axis, find the minimum value of the real number $m$.

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