
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 $f(x)=e^{x}-e^{-x}$ ($x \in \mathbb{R}$) and $e$ is the base of the natural logarithm.

(1) Determine the parity and monotonicity of the function $f(x)$;

(2) Does there exist a real number $t$, such that the inequality $f(x-t)+f(x^{2}-t^{2})\geqslant 0$ holds for all $x$? If it exists, find $t$; if not, explain why.

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