
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 function $f(x)=2^{x}+m2^{1-x}$.
(1) If the function $f(x)$ is an odd function, find the value of the real number $m$;
(2) If the function $f(x)$ is a monotonically increasing function in the interval $(1,+∞)$, find the range of values of the real number $m$;
(3) Determine whether there exists a real number $a$ such that the graph of the function $f(x)$ is symmetric about the point $A(a,0)$. If it exists, find the value of the real number $a$; if not, explain the reason.
Note: The coordinates of the midpoint of points $M(x_{1},y_{1})$, $N(x_{2},y_{2})$ are $(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2})$.

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