
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 $\triangle SIL$, $\measuredangle LIS=60^{\circ}$ and $\measuredangle SIL=75^{\circ}$. $LI=1$. Let $N$ be the midpoint of segment $\overline{LI}$. Point $B$ lies on side $\overline{IL}$ such that $\overline{LB}\perp\overline{SN}$. Extend segment $\overline{IL}$ through $L$ to point $Z$ such that $BZ=ZS$. Find $ZL$ expressed in the form $\frac{p-\sqrt{q}}{r}$, where $p$, $q$, and $r$ are integers.

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