
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 \( P=\left[\frac{1}{2}, 2\right] \) and \( Q=\left\{x \mid ax^{2} - 2x + 2 > 0\right\} \):

(1) If \( P \cap Q \neq \varnothing \), find the range of \( a \);

(2) If \( R=\left\{x \mid ax^{2} - 2x + 2 = 0\right\} \) and \( P \cap R \neq \varnothing \), find the range of \( a \).

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