
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
The odd function $f(x)$, defined on $\mathbb{R}$, has a fundamental period of $2$ and for $x \in (0,1)$, $f(x) =$ $sin(\pi x)$.  
(1) Find the analytic expression for $f(x)$ on the interval $(-1,1)$.  
(2) Determine the monotonicity of $f(x)$ on the interval $(0,1)$, and provide proof.

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