
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
**p10.** Square  $ABCD$  has side length  $n > 1$ . Points  $E$  and  $F$  lie on  $\overline{AB}$  and  $\overline{BC}$  such that  $AE = BF = 1$ . Suppose  $\overline{DE}$  and  $\overline{AF}$  intersect at  $X$  and  $\frac{AX}{XF} = \frac{11}{111}$  . What is  $n$ ?**p11.** Let  $x$  be the positive root of  $x^2 - 10x - 10 = 0$ . Compute  $\frac{1}{20}x^4 - 6x^2 - 45$ .**p12.** Francesca has  $7$  identical marbles and  $5$  distinctly labeled pots. How many ways are there for her to distribute at least one (but not necessarily all) of the marbles into the pots such that at most two pots are nonempty?


PS. You should use hide for answers.

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