
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 that vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are orthogonal unit vectors, $|\overrightarrow{c}|=13$, $\overrightarrow{c}\cdot\overrightarrow{a}=3$, $\overrightarrow{c}\cdot\overrightarrow{b}=4$, find the minimum value of $|\overrightarrow{c}-t_{1}\overrightarrow{a}-t_{2}\overrightarrow{b}|$ for any real numbers $t_{1}$, $t_{2}$.

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