
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 $$\overrightarrow {m}=(f(x), 2\cos x)$$ and $$\overrightarrow {n}=(\sin x+\cos x, 1)$$, and $$\overrightarrow {m} \parallel \overrightarrow {n}$$.
(1) Find the expression for the function $f(x)$.
(2) If the graph of the function $f(x)$ is translated downwards by 1 unit, and then the horizontal coordinate is reduced to half of its original value while keeping the vertical coordinate unchanged, resulting in the graph of function $g(x)$. Find the maximum value of the function $g(x)$ for $$x \in [0, \frac{\pi}{8}]$$ and the corresponding value of $x$.

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