
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
In the diagram, the circle shown has center \(C(6,8)\), radius \(10\), and equation

$$
(x-6)^{2}+(y-8)^{2}=100.
$$

The circle passes through the origin \(O(0,0)\).

(a) Determine the coordinates of the point \(P\) where the circle intersects the \(x\)-axis.

(b) Determine, without justification, the coordinates of the point \(Q\) on the circle that has the maximum \(y\)-coordinate among all points on the circle.

(c) Determine the coordinates of the point \(R\) on the circle so that \(\angle PQR = 90^\circ\).

(d) Determine the coordinates of the two distinct points \(S\) and \(T\) on the circle so that \(\angle PQS = \angle PQT = 45^\circ\).

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