
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
Two circles are centered at the origin. Point $P(10,2)$ is on the larger circle, and point $S(a,a)$ lies on the smaller circle. Given that the distance between points $Q$ and $R$ on the x-axis, where $Q$ and $R$ are on the smaller and larger circles respectively, is $4$, determine the value of $a$.

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