
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 below, we have $\cos \angle TPQ = \frac{4}{5}$. What is $\cos \angle TPS$ if $\angle TPQ$ and $\angle TPS$ are supplementary angles?

[asy]
pair T,P,Q,SS;
SS = (-2,0);
P = (0,0);
Q = (2,0);
T = rotate(aCos(4/5))*(1.5,0);
dot("$S$",SS,S);
dot("$Q$",Q,S);
dot("$T$",T,N);
dot("$P$",P,S);
draw(Q--SS);
draw(P--T);
[/asy]

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