
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 pentagon $PQRST$, sides $QR=RS=ST=1$ unit, $\angle T$ is a right angle, and $m \angle Q = m \angle R = m \angle S = 150^\circ$. The extended lines of $QR$ and $ST$ meet at point $U$. Determine the length of segment $PU$ and find its expression in simplest radical form as $c+\sqrt{d}$ units. What is the value of $c+d$?

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