
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 $p: \sqrt{x-1} \leq 1$, $q: -1 \leq x \leq a$.
$(1)$ If $q$ is a necessary but not sufficient condition for $p$, find the range of real number $a$.
$(2)$ If $a=1$ and at least one of $p$ or $q$ holds true, find the range of $x$.

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