
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
(1) Given the set $P = \{x \mid \frac{1}{2} \leq x \leq 3\}$, the domain of the function $f(x) = \log_2(ax^2 - 2x + 2)$ is denoted as Q. If $P \cap Q = \left[ \frac{1}{2}, \frac{2}{3} \right)$ and $P \cup Q = (-2, 3]$, find the value of the real number $a$.
(2) The function $f(x)$ is defined on $\mathbb{R}$ and satisfies $f(x+3) = f(x)$. When $\frac{1}{2} \leq x \leq 3$, $f(x) = \log_2(ax^2 - 2x + 2)$. If $f(35) = 1$, find the value of the real number $a$.

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