
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
Define an odd function $f(x)$ on $\mathbb{R}$ with the smallest positive period of 4, and for $x \in (0, 2)$, $$f(x)= \frac {3^{x}}{9^{x}+1}.$$  
(1) Find the expression of $f(x)$ on $[-2, 2]$.  
(2) Determine the monotonicity of $f(x)$ on $(0, 2)$ and provide a proof.  
(3) For what value of $\lambda$ does the equation $f(x) = \lambda$ have real solutions on $[-2, 2]$?

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