
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 $f(x) = \begin{cases} \log_{\frac{1}{3}}x & \text{if } x > 0, \\ 2^x & \text{if } x \leq 0, \end{cases}$ find the value of $f\left(\log_{2}\frac{\sqrt{2}}{2}\right) + f(f(9))$ and the range of real numbers $a$ for which $f(f(a)) \leq 1$.

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