
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
Given an isosceles right triangle $ABC \quad\left(CA=CB, \quad \angle C=90^{\circ}\right)$. The segment $AA_{1}$ is the median to the leg $BC$. Point $C_{1}$ is chosen on the hypotenuse $AB$ such that the segments $AA_{1}$ and $CC_{1}$ are perpendicular. Find the ratio $BC_{1}: C_{1}A$.

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