
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 the functions $$f_{1}(x)=e^{|x-2a+1|}$$ and $$f_{2}(x)=e^{|x-a|+1},x\in \mathbb{R}$$.
(1) If $a=2$, find the minimum value of $f(x)=f_{1}(x)+f_{2}(x)$ when $x\in [2,3]$;
(2) If for $x\in [a,+\infty)$, $f_{2}(x) \geq f_{1}(x)$, find the range of values for $a$;
(3) Find the minimum value of the function $$g(x)= \frac {f_{1}(x)+f_{2}(x)}{2}- \frac {|f_{1}(x)-f_{2}(x)|}{2}$$ when $x\in [1,6]$.

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