
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 sets $P = \{x \mid 2x^2 - 3x + 1 \leq 0\}$ and $Q = \{x \mid (x-a)(x-a-1) \leq 0\}$.
$(1)$ If $a = 1$, find the intersection $P \cap Q$;
$(2)$ If $x \in P$ is a sufficient condition for $x \in Q$, find the range of the real number $a$.

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