
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 $\overrightarrow{a}=(3,-1)$, $\overrightarrow{a} \cdot \overrightarrow{b}=-5$, and $\overrightarrow{c}=x \overrightarrow{a}+(1-x) \overrightarrow{b}$.

1. Find the value of the real number $x$ if $\overrightarrow{a} \perp \overrightarrow{c}$.
2. If $|\overrightarrow{b}|= \sqrt {5}$, find the minimum value of $|\overrightarrow{c}|$.

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