
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 any isosceles triangle $PQR$ with $PQ = PR$, the altitude $PS$ bisects the base $QR$ so that $QS = SR$. The lengths of $PQ$ and $PR$ are known to be 13 units each, and $QR$ measures 10 units. Determine the area of $\triangle PQR$.

[asy]
draw((0,0)--(10,0)--(5,12)--cycle,black+linewidth(1));
draw((5,12)--(5,0),black+linewidth(1)+dashed);
draw((5,0)--(5,1)--(4,1)--(4,0)--cycle,black+linewidth(1));
draw((4.5,-2)--(0,-2),black+linewidth(1));
draw((4.5,-2)--(0,-2),EndArrow);
draw((5.5,-2)--(10,-2),black+linewidth(1));
draw((5.5,-2)--(10,-2),EndArrow);
label("$P$",(5,12),N);
label("$Q$",(0,0),SW);
label("$R$",(10,0),SE);
label("$S$",(5,0),S);
label("13",(0,0)--(5,12),NW);
label("13",(5,12)--(10,0),NE);
label("10",(5,-2));
[/asy]

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